%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: 23.dvi %%Pages: 51 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -Z -o 23.ps 23.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.11.02:1650 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 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Fm(\017)49 b Fq(unions)32 b(of)g(arbitrarily)e(man)m(y)j(op)s(en)f (sets)i(are)f(op)s(en.)1918 5251 y(1)p eop %%Page: 2 3 2 2 bop 324 548 a Fq(W)-8 b(e)26 b(will)e(call)h(an)m(y)i(collection)d (of)i(sets)i(on)e Fo(X)34 b Fq(satisfying)25 b(these)j(prop)s(erties)e (a)g(top)s(ology)-8 b(.)324 668 y(In)37 b(the)g(follo)m(wing)d (section,)k(w)m(e)g(also)e(seek)i(to)f(giv)m(e)f(alternativ)m(e)g(w)m (a)m(ys)j(of)d(describing)324 789 y(this)c(imp)s(ortan)m(t)f (collection)f(of)i(sets.)324 1078 y Fl(1.1.1)136 b(De\014ning)45 b(T)-11 b(op)t(ological)46 b(Spaces)324 1262 y Fk(De\014nition)36 b(1.1)49 b Fp(A)31 b Fk(top)s(ological)f(space)h Fp(is)g(a)f(p)-5 b(air)31 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))31 b Fp(c)-5 b(onsisting)29 b(of)i(a)f(set)h Fo(X)324 1383 y Fp(and)j(a)h(family)f Fm(T)61 b Fp(of)34 b(subsets)h(of)g Fo(X)42 b Fp(satisfying)35 b(the)f(fol)5 b(lowing)34 b(c)-5 b(onditions:)320 1586 y(\(T1\))48 b Fm(;)27 b(2)h(T)61 b Fp(and)34 b Fo(X)h Fm(2)29 b(T)320 1789 y Fp(\(T2\))48 b 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Fq(is)32 b(op)s(en)h(in)e(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\))33 b(\(alternativ)m(ely)-8 b(,)31 b Fm(I)7 b Fq(-op)s(en\))61 b Fm(,)1124 4500 y(8)p Fo(x)28 b Fm(2)h Fo(G;)38 b Fm(9)p Fo(r)1598 4515 y Fj(x)1670 4500 y Fo(>)28 b Fq(0)k(suc)m(h)i(that)e(\()p Fo(x)23 b Fm(\000)g Fo(r)2546 4515 y Fj(x)2589 4500 y Fo(;)17 b(x)23 b Fq(+)f Fo(r)2853 4515 y Fj(x)2897 4500 y Fq(\))27 b Fm(\032)h Fo(G:)362 4762 y Fq(\(iii\))46 b(De\014ne)22 b Fm(T)913 4777 y Fh(0)980 4762 y Fq(=)28 b Fm(f;)p Fo(;)17 b(X)8 b Fm(g)53 b Fq(for)32 b(an)m(y)h(set)h(X)e(|)g(kno)m(wn)i(as)f (the)g Fp(trivial)f Fq(or)h Fp(anti-discr)-5 b(ete)31 b Fq(top)s(ology)-8 b(.)364 4965 y(\(iv\))49 b(De\014ne)33 b Fm(D)d Fq(=)e Fm(f)p Fo(G)f Fm(\022)h Fo(X)36 b Fq(:)27 b Fo(G)h Fm(\022)g Fo(X)8 b Fm(g)65 b Fq(|)32 b(kno)m(wn)i(as)f(the)g Fp(discr)-5 b(ete)32 b Fq(top)s(ology)-8 b(.)1918 5251 y(2)p eop %%Page: 3 4 3 3 bop 392 548 a Fq(\(v\))49 b(F)-8 b(or)31 b(an)m(y)h(non-empt)m(y)h (set)f Fo(X)8 b Fq(,)32 b(the)g(family)e Fm(C)k Fq(=)27 b Fm(f)p Fo(G)h Fm(\022)g Fo(X)35 b Fq(:)28 b Fo(G)g Fq(=)f Fm(;)33 b Fq(or)f Fo(X)d Fm(n)20 b Fo(G)32 b Fq(is)568 668 y(\014nite)p Fm(g)g Fq(is)g(a)g(top)s(ology)f(for)h Fo(X)41 b Fq(called)31 b(the)i Fp(c)-5 b(o\014nite)32 b Fq(top)s(ology)-8 b(.)364 872 y(\(vi\))49 b(F)-8 b(or)30 b(an)m(y)i(non-empt)m(y)f(set)h Fo(X)8 b Fq(,)31 b(the)g(family)e Fm(L)e Fq(=)h Fm(f)p Fo(G)f Fm(\022)h Fo(X)36 b Fq(:)28 b Fo(G)f Fq(=)h Fm(;)k Fq(or)g Fo(X)27 b Fm(n)19 b Fo(G)31 b Fq(is)568 992 y(coun)m(table)p Fm(g)h Fq(is)g(a)h(top)s(ology)e(for)h Fo(X)40 b Fq(called)31 b(the)i Fp(c)-5 b(o)g(c)g(ountable)32 b Fq(top)s(ology)-8 b(.)324 1281 y Fl(1.1.2)136 b(Neigh)l(b)t(ourho)t (o)t(ds)324 1466 y Fq(Occasionally)-8 b(,)46 b(argumen)m(ts)f(can)h(b)s (e)f(simpli\014ed)e(when)j(the)f(sets)h(in)m(v)m(olv)m(ed)g(are)f(not) 324 1586 y(\\o)m(v)m(er-describ)s(ed".)78 b(In)44 b(particular,)h(it)e (is)h(sometimes)f(su\016ces)j(to)d(use)i(sets)g(whic)m(h)324 1706 y(con)m(tain)34 b(op)s(en)i(sets)g(but)f(are)g(not)g(necessarily)g (op)s(en.)51 b(W)-8 b(e)36 b(call)d(suc)m(h)j(sets)h(neigh)m(b)s(or-) 324 1827 y(ho)s(o)s(ds.)324 2055 y Fk(De\014nition)f(1.2)49 b Fp(Given)37 b(a)h(top)-5 b(olo)g(gic)g(al)36 b(sp)-5 b(ac)g(e)37 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))38 b Fp(with)f Fo(x)c Fm(2)g Fo(X)8 b Fp(,)38 b(then)g Fo(N)43 b Fm(\022)33 b Fo(X)324 2176 y Fp(is)h(said)h(to)g(b)-5 b(e)34 b(a)h(\()p Fm(T)25 b Fp(\)-)p Fk(neigh)m(b)s(ourho)s(o)s(d)36 b Fp(of)f Fo(x)28 b Fm(,)f(9)35 b Fp(op)-5 b(en)34 b(set)h Fo(G)g Fp(with)g Fo(x)28 b Fm(2)g Fo(G)g Fm(\022)g Fo(N)10 b Fp(.)324 2404 y Fq(It)36 b(follo)m(ws)f(then)h(that)g(a)g(set)h Fo(U)44 b Fm(\022)34 b Fo(X)44 b Fq(is)35 b(op)s(en)h(i\013)f(for)h(ev) m(ery)i Fo(x)c Fm(2)g Fo(U)10 b Fq(,)37 b(there)g(exists)g(a)324 2524 y(neigh)m(b)s(ourho)s(o)s(d)31 b Fo(N)1068 2539 y Fj(x)1145 2524 y Fq(of)h Fo(x)h Fq(con)m(tained)g(in)e Fo(U)10 b Fq(.)45 b(\(Chec)m(k)34 b(this!\))324 2728 y Fk(Lemma)j(1.1)49 b Fp(L)-5 b(et)30 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))30 b Fp(b)-5 b(e)29 b(a)h(top)-5 b(olo)g(gic)g(al)28 b(sp)-5 b(ac)g(e)29 b(and,)h(for)g(e)-5 b(ach)29 b Fo(x)f Fm(2)g Fo(X)8 b Fp(,)30 b(let)g Fm(N)15 b Fq(\()p Fo(x)p Fq(\))324 2848 y Fp(b)-5 b(e)34 b(the)h(family)g(of)f(neighb)-5 b(ourho)g(o)g(ds)33 b(of)i Fo(x)p Fp(.)45 b(Then)409 3051 y(\(i\))k Fo(U)38 b Fm(2)28 b(N)15 b Fq(\()p Fo(x)p Fq(\))27 b Fm(\))h Fo(x)g Fm(2)g Fo(U:)379 3255 y Fp(\(ii\))49 b Fm(N)15 b Fq(\()p Fo(x)p Fq(\))34 b Fp(is)h(close)-5 b(d)34 b(under)h(\014nite)f(interse)-5 b(ctions.)350 3458 y(\(iii\))48 b Fo(U)38 b Fm(2)28 b(N)15 b Fq(\()p Fo(x)p Fq(\))35 b Fp(and)f Fo(U)k Fm(\022)28 b Fo(V)50 b Fm(\))27 b Fo(V)49 b Fm(2)28 b(N)15 b Fq(\()p Fo(x)p Fq(\))p Fo(:)364 3662 y Fp(\(iv\))49 b Fo(U)44 b Fm(2)34 b(N)15 b Fq(\()p Fo(x)p Fq(\))33 b Fm(\))g(9)p Fo(W)48 b Fm(2)34 b(N)15 b Fq(\()p Fo(x)p Fq(\))38 b Fp(such)g(that)g Fo(W)47 b Fm(\022)35 b Fo(U)48 b Fp(and)38 b Fo(W)47 b Fm(2)34 b(N)15 b Fq(\()p Fo(y)t Fq(\))36 b Fp(for)i(e)-5 b(ach)568 3782 y Fo(y)30 b Fm(2)f Fo(W)m(:)324 3985 y Fq(Pro)s(of)p 324 3998 235 4 v 32 w(Exercise!)324 4245 y Fk(Examples)416 4430 y Fq(\(i\))48 b(Let)34 b Fo(x)e Fm(2)f Fo(X)8 b Fq(,)35 b(and)f(de\014ne)i Fm(T)1608 4445 y Fj(x)1683 4430 y Fq(=)31 b Fm(f;)p Fo(;)17 b Fm(f)p Fo(x)p Fm(g)p Fo(;)g(X)8 b Fm(g)p Fq(.)48 b(Then)35 b Fm(T)2657 4445 y Fj(x)2736 4430 y Fq(is)f(a)g(top)s(ology)f(for)h Fo(X)568 4550 y Fq(and)f Fo(V)51 b Fm(\022)30 b Fo(X)41 b Fq(is)33 b(a)g(neigh)m(b)s(ourho)s(o)s(d)g(of)g Fo(x)h Fq(i\013)e Fo(x)e Fm(2)f Fo(V)22 b Fq(.)46 b(Ho)m(w)m(ev)m(er,)36 b(the)e(only)f(nhd)568 4671 y(of)f Fo(y)f Fm(2)d Fo(X)40 b Fq(where)34 b Fo(y)d Fm(6)p Fq(=)c Fo(x)33 b Fq(is)f Fo(X)40 b Fq(itself)1918 5251 y(3)p eop %%Page: 4 5 4 4 bop 389 548 a Fq(\(ii\))47 b(Let)32 b Fo(x)d Fm(2)f Fo(X)40 b Fq(and)33 b(de\014ne)g(a)g(top)s(ology)e Fm(I)7 b Fq(\()p Fo(x)p Fq(\))33 b(for)f Fo(X)41 b Fq(as)32 b(follo)m(ws:)1392 710 y Fm(I)7 b Fq(\()p Fo(x)p Fq(\))28 b(=)g Fm(f)p Fo(G)f Fm(\022)h Fo(X)36 b Fq(:)28 b Fo(x)g Fm(2)g Fo(G)p Fm(g)22 b([)g(f;g)p Fo(:)568 872 y Fq(Note)32 b(here)i(that)e Fp(every)h Fq(nhd)g(of)f(a)g(p)s(oin)m(t)g(in)g Fo(X)40 b Fq(is)32 b(op)s(en.)362 1059 y(\(iii\))46 b(Let)32 b Fo(x)d Fm(2)f Fo(X)40 b Fq(and)33 b(de\014ne)g(a)g(top)s(ology)e Fm(E)9 b Fq(\()p Fo(x)p Fq(\))32 b(for)g Fo(X)40 b Fq(as)33 b(follo)m(ws:)1372 1222 y Fm(E)9 b Fq(\()p Fo(x)p Fq(\))28 b(=)f Fm(f)p Fo(G)h Fm(\022)g Fo(X)35 b Fq(:)28 b Fo(x)g Fm(62)g Fo(G)p Fm(g)22 b([)h(f)p Fo(X)8 b Fm(g)p Fo(:)568 1384 y Fq(Note)41 b(here)h(that)f Fm(f)p Fo(y)t Fm(g)e Fq(is)i(op)s(en)g(for)g(ev)m(ery)i Fo(y)i Fm(6)p Fq(=)d Fo(x)g Fq(in)e Fo(X)8 b Fq(,)43 b(that)e Fm(f)p Fo(x;)17 b(y)t Fm(g)40 b Fq(is)h Fp(not)568 1504 y Fq(op)s(en,)33 b(is)f(not)g(a)g(nhd)h(of)g Fo(x)f Fq(y)m(et)i Fp(is)e Fq(a)h(nhd)g(of)f Fo(y)t Fq(.)568 1658 y(In)h(fact,)f(the)h(only)f(nhd) h(of)f Fo(x)h Fq(is)f Fo(X)8 b Fq(.)324 1938 y Fl(1.1.3)136 b(Bases)45 b(and)g(Subbases)324 2123 y Fq(It)34 b(often)g(happ)s(ens)h (that)e(the)i(op)s(en)f(sets)h(of)f(a)f(space)i(can)g(b)s(e)f(v)m(ery)h (complicated)d(and)324 2243 y Fp(yet)37 b Fq(they)g(can)g(all)d(b)s(e)j (describ)s(ed)g(using)f(a)h(selection)f(of)g(fairly)e(simple)h(sp)s (ecial)h(ones.)324 2364 y(When)d(this)e(happ)s(ens,)i(the)g(set)f(of)g (simple)e(op)s(en)i(sets)h(is)f(called)f(a)g Fk(base)i Fq(or)f Fk(subbase)324 2484 y Fq(\(dep)s(ending)c(on)g(ho)m(w)g(the)h (description)e(is)h(to)f(b)s(e)h(done\).)43 b(In)28 b(addition,)f(it)g (is)h(fortunate)324 2604 y(that)c(man)m(y)g(top)s(ological)d(concepts)k (can)g(b)s(e)f(c)m(haracterized)h(in)f(terms)g(of)g(these)h(simpler)324 2725 y(base)33 b(or)f(subbase)j(elemen)m(ts.)324 2879 y Fk(De\014nition)h(1.3)49 b Fp(L)-5 b(et)36 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))36 b Fp(b)-5 b(e)35 b(a)g(top)-5 b(olo)g(gic)g(al)35 b(sp)-5 b(ac)g(e.)46 b(A)35 b(family)g Fm(B)e(\022)c(T)61 b Fp(is)36 b(c)-5 b(al)5 b(le)-5 b(d)324 2999 y(a)38 b Fk(base)44 b(for)e Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))39 b Fp(if)f(and)g(only)h(if)g(every)f(non-empty)h (op)-5 b(en)38 b(subset)h(of)f Fo(X)47 b Fp(c)-5 b(an)38 b(b)-5 b(e)324 3119 y(r)g(epr)g(esente)g(d)34 b(as)g(a)h(union)g(of)f (a)h(subfamily)f(of)h Fm(B)s Fp(.)324 3273 y Fq(It)h(is)f(easily)g(v)m (eri\014ed)i(that)e Fm(B)i(\022)d(T)61 b Fq(is)36 b(a)f(base)i(for)e (\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))35 b(if)g(and)h(only)f(if)g(whenev) m(er)324 3394 y Fo(x)28 b Fm(2)g Fo(G)g Fm(2)g(T)d Fo(;)17 b Fm(9)p Fo(B)33 b Fm(2)28 b(B)36 b Fq(suc)m(h)e(that)f Fo(x)28 b Fm(2)g Fo(B)33 b Fm(\022)28 b Fo(G)p Fq(.)324 3514 y(Clearly)-8 b(,)31 b(a)i(top)s(ological)c(space)k(can)g(ha)m(v)m (e)h(man)m(y)f(bases.)324 3676 y Fk(Lemma)k(1.2)49 b Fp(If)34 b Fm(B)39 b Fp(is)34 b(a)h(family)f(of)h(subsets)g(of)f(a)h (set)g Fo(X)42 b Fp(such)35 b(that)321 3830 y(\(B1\))48 b(for)37 b(any)h Fo(B)990 3845 y Fh(1)1030 3830 y Fo(;)17 b(B)1148 3845 y Fh(2)1221 3830 y Fm(2)34 b(B)41 b Fp(and)d(every)g(p)-5 b(oint)37 b Fo(x)d Fm(2)g Fo(B)2393 3845 y Fh(1)2457 3830 y Fm(\\)25 b Fo(B)2622 3845 y Fh(2)2662 3830 y Fp(,)38 b(ther)-5 b(e)38 b(exists)g Fo(B)3320 3845 y Fh(3)3393 3830 y Fm(2)c(B)568 3951 y Fp(with)g Fo(x)28 b Fm(2)h Fo(B)1031 3966 y Fh(3)1098 3951 y Fm(\022)f Fo(B)1277 3966 y Fh(1)1339 3951 y Fm(\\)22 b Fo(B)1501 3966 y Fh(2)1541 3951 y Fp(,)34 b(and)321 4138 y(\(B2\))48 b(for)34 b(every)h Fo(x)28 b Fm(2)g Fo(X)8 b Fp(,)35 b(ther)-5 b(e)35 b(exists)f Fo(B)f Fm(2)28 b(B)38 b Fp(such)d(that)g Fo(x)28 b Fm(2)h Fo(B)5 b Fp(,)324 4291 y(then)34 b Fm(B)39 b Fp(is)34 b(a)h(b)-5 b(ase)34 b(for)h(a)g(unique)f(top)-5 b(olo)g(gy)35 b(on)g Fo(X)8 b Fp(.)324 4412 y(Conversely,)28 b(any)f(b)-5 b(ase)27 b Fm(B)k Fp(for)c(a)g(top)-5 b(olo)g(gic)g(al)27 b(sp)-5 b(ac)g(e)26 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))h Fp(satis\014es)g(\(B1\))g(and)g(\(B2\).)324 4574 y Fq(Pro)s(of)p 324 4587 235 4 v 32 w(\(Exercise!\))324 4736 y Fk(De\014nition)36 b(1.4)49 b Fp(L)-5 b(et)36 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))36 b Fp(b)-5 b(e)35 b(a)g(top)-5 b(olo)g(gic)g(al)35 b(sp)-5 b(ac)g(e.)46 b(A)36 b(family)f Fm(S)i(\022)29 b(T)61 b Fp(is)36 b(c)-5 b(al)5 b(le)-5 b(d)324 4857 y(a)46 b Fk(subbase)54 b(for)d Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))46 b Fp(if)h(and)e(only)i(if)f (the)g(family)g(of)g(al)5 b(l)46 b(\014nite)g(interse)-5 b(ctions)324 4977 y Fm(\\)390 4936 y Fj(k)390 5002 y(i)p Fh(=1)509 4977 y Fo(U)575 4992 y Fj(i)603 4977 y Fp(,)35 b(wher)-5 b(e)34 b Fo(U)1009 4992 y Fj(i)1065 4977 y Fm(2)28 b(S)43 b Fp(for)34 b Fo(i)28 b Fq(=)g(1)p Fo(;)17 b Fq(2)p Fo(;)g(:)g(:)g(:)e(;)i(k)38 b Fp(is)c(a)h(b)-5 b(ase)34 b(for)h Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))p Fp(.)1918 5251 y Fq(4)p eop %%Page: 5 6 5 5 bop 324 548 a Fk(Examples)416 733 y Fq(\(i\))48 b(In)39 b(an)m(y)g(metric)f(space)i(\()p Fo(X)r(;)17 b(d)p Fq(\),)40 b Fm(f)p Fo(B)1908 748 y Fj(r)1946 733 y Fq(\()p Fo(x)p Fq(\))e(:)h Fo(x)f Fm(2)h Fo(X)r(;)34 b(r)41 b(>)d Fq(0)p Fm(g)g Fq(forms)g(a)h(base)g(for)568 853 y(the)33 b(induced)g(metric)e (top)s(ology)g Fm(T)1857 868 y Fj(d)1931 853 y Fq(on)h Fo(X)8 b Fq(.)389 1056 y(\(ii\))47 b(F)-8 b(or)47 b(the)i(real)e(line)g Fo(R)i Fq(with)f(its)f(usual)h(\(Euclidean\))g(top)s(ology)-8 b(,)50 b(the)f(family)568 1177 y Fm(f)p Fq(\()p Fo(a;)17 b(b)p Fq(\))27 b(:)h Fo(a;)17 b(b)28 b Fm(2)g Fo(Q;)34 b(a)28 b(<)f(b)p Fm(g)33 b Fq(is)f(a)h(base.)362 1380 y(\(iii\))46 b(F)-8 b(or)31 b(an)i(arbitrary)e(set)j Fo(X)8 b Fq(,)32 b(the)h(family)d Fm(ff)p Fo(x)p Fm(g)e Fq(:)g Fo(x)g Fm(2)g Fo(X)8 b Fm(g)32 b Fq(is)g(a)g(base)i(for)e(\()p Fo(X)r(;)17 b Fm(D)s Fq(\).)364 1584 y(\(iv\))49 b(The)30 b(family)c(of)j(all)e(`semi-in\014nite')g(op)s(en)j(in)m(terv)-5 b(als)28 b(\()p Fo(a;)17 b Fm(1)p Fq(\))29 b(and)g(\()p Fm(\0001)p Fo(;)17 b(b)p Fq(\))29 b(in)g Fo(R)568 1704 y Fq(is)j(a)g(subbase)i(for)e(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\).)324 1993 y Fl(1.1.4)136 b(Generating)46 b(T)-11 b(op)t(ologies)324 2178 y Fq(F)j(rom)30 b(the)i(ab)s(o)m(v)m(e)g (examples,)g(it)f(follo)m(ws)f(that)i(for)f(a)g(set)h Fo(X)40 b Fq(one)32 b(can)g(select)g(in)f(man)m(y)324 2298 y(di\013eren)m(t)j(w)m(a)m(ys)i(a)e(family)d Fm(T)60 b Fq(suc)m(h)35 b(that)f(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))34 b(is)f(a)h(top)s(ological)d(space.)49 b(If)34 b Fm(T)3331 2313 y Fh(1)3404 2298 y Fq(and)324 2418 y Fm(T)378 2433 y Fh(2)447 2418 y Fq(are)c(t)m(w)m(o)h(top)s(ologies)c (for)j Fo(X)37 b Fq(and)30 b Fm(T)1750 2433 y Fh(2)1818 2418 y Fm(\022)e(T)1977 2433 y Fh(1)2017 2418 y Fq(,)i(then)h(w)m(e)f (sa)m(y)h(that)f(the)g(top)s(ology)f Fm(T)3427 2433 y Fh(1)3496 2418 y Fq(is)324 2539 y(\014ner)36 b(than)g(the)g(top)s (ology)e Fm(T)1409 2554 y Fh(2)1449 2539 y Fq(,)i(or)g(that)f Fm(T)1903 2554 y Fh(2)1979 2539 y Fq(is)g(coarser)h(than)g(the)g(top)s (ology)e Fm(T)3274 2554 y Fh(1)3314 2539 y Fq(.)53 b(The)324 2659 y(discrete)32 b(top)s(ology)e(for)h Fo(X)39 b Fq(is)31 b(the)h(\014nest)h(one;)f(the)g(trivial)d(top)s(ology)h(is)h(the)g (coarsest.)324 2780 y(If)38 b Fo(X)46 b Fq(is)37 b(an)h(arbitrary)f (in\014nite)g(set)i(with)f(distinct)f(p)s(oin)m(ts)h Fo(x)g Fq(and)h Fo(y)t Fq(,)f(then)h(one)f(can)324 2900 y(readily)25 b(v)m(erify)j(that)e(the)h(top)s(ologies)e Fm(I)7 b Fq(\()p Fo(x)p Fq(\))28 b(and)f Fm(I)7 b Fq(\()p Fo(y)t Fq(\))26 b(are)h(incomparable)e(i.e.)41 b(neither)324 3020 y(is)32 b(\014ner)h(than)g(the)g(other.)324 3141 y(By)j Fp(gener)-5 b(ating)36 b(a)i(top)-5 b(olo)g(gy)35 b Fq(for)g Fo(X)8 b Fq(,)36 b(w)m(e)g(mean)f(selecting)g(a)h(family)d Fm(T)61 b Fq(of)35 b(subsets)i(of)324 3261 y Fo(X)44 b Fq(whic)m(h)37 b(satis\014es)g(conditions)f(\(T1\){\(T3\).)55 b(Often)36 b(it)g(is)g(more)f(con)m(v)m(enien)m(t)k Fp(not)d Fq(to)324 3381 y(describ)s(e)f(the)g(family)d Fm(T)60 b Fq(of)34 b(op)s(en)h(sets)g(directly)-8 b(.)49 b(The)36 b(concept)f(of)f(a)g(base)i(o\013ers)f(an)324 3502 y(alternativ)m(e)d (metho)s(d)g(of)g(generating)g(top)s(ologies.)324 3762 y Fk(Examples)469 3946 y Fm(\017)49 b Fq([Sorgenfrey)37 b(line])e(Giv)m(en)h(the)h(real)f(n)m(um)m(b)s(ers)h Fo(R)q Fq(,)g(let)f Fm(B)k Fq(b)s(e)c(the)h(family)d(of)i(all)568 4067 y(in)m(terv)-5 b(als)30 b([)p Fo(x;)17 b(r)s Fq(\))31 b(where)h Fo(x)p Fq(,)p Fo(r)f Fm(2)d Fo(R)q Fq(,)j Fo(x)d(<)g(r)33 b Fq(and)f Fo(r)h Fq(is)e(rational.)40 b(One)31 b(can)h(readily)568 4187 y(c)m(hec)m(k)g(that)d Fm(B)34 b Fq(has)c(prop)s(erties)g (\(B1\){\(B2\).)41 b(The)31 b(space)g Fo(R)2797 4202 y Fj(s)2834 4187 y Fq(,)g(generated)f(b)m(y)h Fm(B)s Fq(,)568 4307 y(is)e(called)g(the)i Fp(Sor)-5 b(genfr)g(ey)32 b(line)d Fq(and)i(has)f(man)m(y)g(in)m(teresting)g(prop)s(erties.)42 b(Note)568 4428 y(that)c(the)h(Sorgenfrey)g(top)s(ology)e(is)i(\014ner) g(than)f(the)h(Euclidean)g(top)s(ology)e(on)568 4548 y Fo(R)q Fq(.)43 b(\(Chec)m(k!\))469 4752 y Fm(\017)49 b Fq([Niem)m(ytzki)30 b(plane])g(Let)h Fo(L)g Fq(denote)h(the)f(closed) g(upp)s(er)g(half-plane.)41 b(W)-8 b(e)31 b(de\014ne)568 4872 y(a)h(top)s(ology)f(for)h Fo(L)h Fq(b)m(y)g(declaring)f(the)h (basic)f(op)s(en)h(sets)h(to)e(b)s(e)h(the)g(follo)m(wing:)1918 5251 y(5)p eop %%Page: 6 7 6 6 bop 622 548 a Fq(\(I\))49 b(the)33 b(\(Euclidean\))f(op)s(en)h (discs)g(in)f(the)h(upp)s(er)g(half-plane;)585 710 y(\(I)s(I\))48 b(the)28 b(\(Euclidean\))e(op)s(en)h(discs)g(tangen)m(t)h(to)e(the)i (`edge')f(of)g(the)g Fo(L)p Fq(,)h(together)782 830 y(with)k(the)h(p)s (oin)m(t)f(of)g(tangency)-8 b(.)568 1034 y Fk(Note)32 b Fq(If)g Fo(y)984 1049 y Fj(n)1058 1034 y Fm(!)c Fo(y)35 b Fq(in)d Fo(L)p Fq(,)h(then)631 1237 y(\(i\))47 b Fo(y)36 b Fq(not)c(on)h(`edge':)44 b(same)33 b(as)f(Euclidean)g(con)m(v)m (ergence.)603 1399 y(\(ii\))47 b Fo(y)35 b Fq(on)c(the)h(`edge':)44 b(same)31 b(as)g(Euclidean,)h(but)f Fo(y)2544 1414 y Fj(n)2622 1399 y Fq(m)m(ust)h(approac)m(h)f Fo(y)k Fq(from)782 1519 y(`inside'.)43 b(Th)m(us,)35 b(for)d(example,)g Fo(y)2032 1534 y Fj(n)2106 1519 y Fq(=)c(\()2262 1480 y Fh(1)p 2258 1496 43 4 v 2258 1554 a Fj(n)2310 1519 y Fo(;)17 b Fq(0\))27 b Fm(6!)h Fq(\(0)p Fo(;)17 b Fq(0\)!)324 1808 y Fl(1.1.5)136 b(New)45 b(Spaces)g(from)g(Old)324 1993 y Fq(A)33 b(subset)h(of)e(a)h(top)s(ological)c(space)k(inherits)f (a)h(top)s(ology)e(of)h(its)g(o)m(wn,)i(in)e(an)g(ob)m(vious)324 2113 y(w)m(a)m(y:)324 2317 y Fk(De\014nition)k(1.5)49 b Fp(Given)31 b(a)g(top)-5 b(olo)g(gic)g(al)30 b(sp)-5 b(ac)g(e)30 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))31 b Fp(with)g Fo(A)d Fm(\022)g Fo(X)8 b Fp(,)31 b(then)g(the)g(fam-)324 2437 y(ily)44 b Fm(T)524 2452 y Fj(A)627 2437 y Fq(=)h Fm(f)p Fo(A)29 b Fm(\\)h Fo(G)45 b Fq(:)h Fo(G)f Fm(2)h(T)26 b(g)44 b Fp(is)g(a)g(top)-5 b(olo)g(gy)45 b(for)f Fo(A)p Fp(,)j(c)-5 b(al)5 b(le)-5 b(d)43 b(the)i Fk(subspace)h Fp(\(or)324 2557 y Fk(relativ)m(e)30 b Fp(or)i Fk(induced)p Fp(\))f(top)-5 b(olo)g(gy)32 b(for)f Fo(A)p Fp(.)44 b Fq(\()p Fo(A;)17 b Fm(T)2174 2572 y Fj(A)2231 2557 y Fq(\))32 b Fp(is)f(c)-5 b(al)5 b(le)-5 b(d)31 b(a)h(subsp)-5 b(ac)g(e)31 b(of)g Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))p Fp(.)324 2761 y(Example)324 2881 y Fq(The)44 b(in)m(terv)-5 b(al)41 b Fo(I)53 b Fq(=)45 b([0)p Fo(;)17 b Fq(1])42 b(with)g(its)h(natural)e(\(Euclidean\))h(top)s(ology)g(is)g(a)h (\(closed\))324 3002 y(subspace)34 b(of)e(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\).)324 3122 y Fp(Warning)p Fq(:)40 b(Although)25 b(this)h(de\014nition,)g(and)h(sev)m(eral)f(of)g(the)h(results)f(whic)m (h)h(\015o)m(w)g(from)324 3242 y(it,)34 b(ma)m(y)g(suggest)i(that)e (subspaces)j(in)d(general)g(top)s(ology)f(are)h(going)g(to)g(b)s(e)h (`easy')g(in)324 3363 y(the)i(sense)i(that)d(a)h(lot)f(of)g(the)h (structure)h(just)g(gets)f(traced)g(on)m(to)g(the)g(subset,)j(there)324 3483 y(is)29 b(unfortunately)g(a)h(ric)m(h)f(source)i(of)e(mistak)m(es) h(here)h(also:)41 b(b)s(ecause)31 b(w)m(e)f(are)g(handling)324 3604 y(t)m(w)m(o)40 b(top)s(ologies)d(at)i(once.)65 b(When)41 b(w)m(e)f(insp)s(ect)g(a)f(subset)i Fo(B)k Fq(of)39 b Fo(A)p Fq(,)i(and)f(refer)f(to)h(it)324 3724 y(as)35 b('op)s(en')h(\(or)f('closed',)h(or)f(a)g('neigh)m(b)s(ourho)s(o)s(d')g (of)f(some)h(p)s(oin)m(t)g(p)g(.)16 b(.)g(.)g(.)53 b(\))e(w)m(e)36 b(m)m(ust)324 3844 y(b)s(e)31 b(exceedingly)h(careful)e(as)i(to)e Fp(which)38 b Fq(top)s(ology)29 b(is)i(in)m(tended.)44 b(F)-8 b(or)30 b(instance,)i(in)e(the)324 3965 y(previous)36 b(example,)h([0)p Fo(;)17 b Fq(1])36 b(itself)f(is)g(op)s(en)i(in)e (the)i(subspace)h(top)s(ology)d(on)h Fo(I)44 b Fq(but,)37 b(of)324 4085 y(course,)i(not)e(in)f(the)h('bac)m(kground')h(top)s (ology)e(of)g Fo(R)q Fq(.)57 b(In)37 b(suc)m(h)h(circumstances,)h(it)d (is)324 4205 y(advisable)41 b(to)h Fp(sp)-5 b(e)g(cify)50 b Fq(the)42 b(top)s(ology)e(b)s(eing)h(used)i(eac)m(h)g(time)e(b)m(y)h (sa)m(ying)g Fm(T)26 b Fq(-op)s(en,)324 4326 y Fm(T)378 4341 y Fj(A)435 4326 y Fq(-op)s(en,)32 b(and)h(so)g(on.)324 4659 y Fn(1.2)160 b(Closed)53 b(sets)g(and)g(Closure)324 4878 y Fq(Just)27 b(as)g(man)m(y)g(concepts)i(in)d(metric)g(spaces)i(w) m(ere)h(describ)s(ed)e(in)f(terms)h(of)g(basic)f(op)s(en)324 4998 y(sets,)33 b(y)m(et)f(others)g(w)m(ere)g(c)m(haracterized)g(in)f (terms)g(of)g(closed)g(sets.)45 b(In)31 b(this)g(section)h(w)m(e)1918 5251 y(6)p eop %%Page: 7 8 7 7 bop 469 548 a Fm(\017)49 b Fq(de\014ne)33 b(closed)g(sets)h(in)e(a) g(general)g(top)s(ological)d(space)34 b(and)469 749 y Fm(\017)49 b Fq(examine)32 b(the)h(related)f(notion)f(of)h(the)h (closure)g(of)f(a)h(giv)m(en)f(set.)324 1036 y Fl(1.2.1)136 b(Closed)45 b(Sets)324 1221 y Fk(De\014nition)36 b(1.6)49 b Fp(Given)28 b(a)g(top)-5 b(olo)g(gic)g(al)27 b(sp)-5 b(ac)g(e)27 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))i Fp(with)g Fo(F)41 b Fm(\022)28 b Fo(X)8 b Fp(,)30 b(then)e Fo(F)41 b Fp(is)28 b(said)324 1341 y(to)35 b(b)-5 b(e)34 b Fm(T)26 b Fp(-close)-5 b(d)34 b(i\013)g(its)h(c)-5 b(omplement)34 b Fo(X)c Fm(n)22 b Fo(F)48 b Fp(is)35 b Fm(T)25 b Fp(-op)-5 b(en.)324 1558 y Fq(F)d(rom)25 b(De)h(Morgan's)g(La)m(ws)h(and)g(prop)s(erties)f(\(T1\){\(T3\))g(of)g (op)s(en)g(sets,)j(w)m(e)e(infer)f(that)324 1679 y(the)33 b(family)d Fm(F)42 b Fq(of)32 b(closed)h(sets)h(of)e(a)g(space)i(has)f (the)g(follo)m(wing)c(prop)s(erties:)331 1874 y(\(F1\))48 b Fo(X)35 b Fm(2)28 b(F)42 b Fq(and)33 b Fm(;)27 b(2)h(F)331 2075 y Fq(\(F2\))48 b Fm(F)42 b Fq(is)32 b(closed)h(under)g(\014nite)f (union)331 2275 y(\(F3\))48 b Fm(F)42 b Fq(is)32 b(closed)h(under)g (arbitrary)f(in)m(tersection.)324 2471 y(Sets)46 b(whic)m(h)f(are)g (sim)m(ultaneously)e(closed)i Fp(and)g Fq(op)s(en)g(in)f(a)g(top)s (ological)e(space)k(are)324 2591 y(sometimes)41 b(referred)i(to)f(as)h Fk(clop)s(en)f Fq(sets.)73 b(F)-8 b(or)42 b(example,)i(mem)m(b)s(ers)e (of)g(the)h(base)324 2711 y Fm(B)35 b Fq(=)c Fm(f)p Fq([)p Fo(x;)17 b(r)s Fq(\))31 b(:)g Fo(x;)j(r)g Fm(2)e Fo(R)q(;)h(x)f(<)f(r)m (;)j(r)h Fq(rational)30 b Fm(g)35 b Fq(for)f(the)h(Sorgenfrey)h(line)d (are)i(clop)s(en)324 2832 y(with)f(resp)s(ect)i(to)e(the)h(top)s(ology) e(generated)i(b)m(y)g Fm(B)s Fq(.)50 b(Indeed,)36 b(for)e(the)h (discrete)g(space)324 2952 y(\()p Fo(X)r(;)17 b Fm(D)s Fq(\),)32 b Fp(every)g Fq(subset)j(is)d(clop)s(en.)324 3239 y Fl(1.2.2)136 b(Closure)45 b(of)h(Sets)324 3424 y Fk(De\014nition)36 b(1.7)49 b Fp(If)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))35 b Fp(is)f(a)h(top)-5 b(olo)g(gic)g(al)34 b(sp)-5 b(ac)g(e)34 b(and)g Fo(A)28 b Fm(\022)g Fo(X)8 b Fp(,)34 b(then)p 1051 3573 74 4 v 1051 3651 a Fo(A)1124 3593 y Fg(T)1212 3651 y Fq(=)28 b Fm(\\f)p Fo(F)41 b Fm(\022)28 b Fo(X)36 b Fq(:)28 b Fo(A)f Fm(\022)i Fo(F)48 b Fp(and)34 b Fo(F)49 b Fp(is)35 b(close)-5 b(d)o Fm(g)324 3862 y Fp(is)34 b(c)-5 b(al)5 b(le)-5 b(d)34 b(the)h Fm(T)26 b Fk(-closure)34 b Fp(of)h Fo(A)p Fp(.)324 4090 y Fq(Eviden)m(tly)-8 b(,)p 793 4012 V 40 w Fo(A)866 4032 y Fg(T)964 4090 y Fq(\(or)p 1127 4012 V 38 w Fo(A)38 b Fq(when)i(there)f(is)e(no)h(danger)h(of)e(am)m(biguit)m(y\))g(is)h (the)g(smallest)324 4210 y(closed)33 b(subset)h(of)e Fo(X)40 b Fq(whic)m(h)33 b(con)m(tains)g Fo(A)p Fq(.)43 b(Note)33 b(that)f Fo(A)h Fq(is)f(closed)h Fm(,)27 b Fo(A)h Fq(=)p 3219 4132 V 28 w Fo(A)p Fq(.)324 4406 y Fk(Lemma)37 b(1.3)49 b Fp(If)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))35 b Fp(is)f(a)h(top)-5 b(olo)g(gic)g(al)34 b(sp)-5 b(ac)g(e)34 b(with)g Fo(A)p Fp(,)h Fo(B)e Fm(\022)28 b Fo(X)8 b Fp(,)35 b(then)409 4601 y(\(i\))568 4569 y Fq(\026)568 4601 y Fm(;)27 b Fq(=)h Fm(;)379 4801 y Fp(\(ii\))49 b Fo(A)27 b Fm(\022)799 4776 y Fq(\026)774 4801 y Fo(A)350 5002 y Fp(\(iii\))580 4964 y Fq(\026)593 4977 y(\026)568 5002 y Fo(A)g Fq(=)798 4977 y(\026)772 5002 y Fo(A)1918 5251 y Fq(7)p eop %%Page: 8 9 8 8 bop 364 548 a Fp(\(iv\))p 568 470 263 4 v 49 w Fo(A)22 b Fm([)g Fo(B)33 b Fq(=)988 523 y(\026)962 548 y Fo(A)22 b Fm([)1169 523 y Fq(\026)1146 548 y Fo(B)5 b Fp(.)324 744 y Fq(Pro)s(of)p 324 757 235 4 v 32 w(Exercise!)324 962 y Fk(Theorem)37 b(1.1)49 b Fp(Given)44 b(a)g(top)-5 b(olo)g(gic)g(al)43 b(sp)-5 b(ac)g(e)44 b(with)g Fo(A)i Fm(\022)f Fo(X)8 b Fp(,)47 b(then)d Fo(x)i Fm(2)3215 937 y Fq(\026)3189 962 y Fo(A)f Fp(i\013)f(for)324 1083 y Fq(eac)m(h)36 b Fp(nhd)e Fo(U)45 b Fp(of)35 b Fo(x)p Fp(,)g Fo(U)e Fm(\\)23 b Fo(A)k Fm(6)p Fq(=)h Fm(;)p Fp(.)324 1301 y Fq(Pro)s(of)p 324 1314 V 392 1497 a Fm(\))p Fq(:)49 b(Let)42 b Fo(x)j Fm(2)989 1472 y Fq(\026)963 1497 y Fo(A)d Fq(and)h(let)f Fo(U)53 b Fq(b)s(e)42 b(a)g(nhd)h(of)f Fo(x)p Fq(;)48 b(then)43 b(there)g(exists)g(op)s(en)g Fo(G)f Fq(with)568 1617 y Fo(x)28 b Fm(2)g Fo(G)f Fm(\022)i Fo(U)10 b Fq(.)40 b(If)23 b Fo(U)12 b Fm(\\)q Fo(A)28 b Fq(=)f Fm(;)p Fq(,)d(then)f Fo(G)q Fm(\\)q Fo(A)29 b Fq(=)e Fm(;)22 b Fq(and)h(so)f Fo(A)28 b Fm(\022)g Fo(X)9 b Fm(n)q Fo(G)28 b Fm(\))3164 1592 y Fq(\026)3138 1617 y Fo(A)g Fm(\022)g Fo(X)9 b Fm(n)q Fo(G)568 1737 y Fq(whence)24 b Fo(x)k Fm(2)g Fo(X)9 b Fm(n)q Fo(G)p Fq(,)24 b(thereb)m(y)g(con)m(tradicting)d(the)i(assumption)f(that)g Fo(U)12 b Fm(\\)q Fo(A)28 b Fq(=)g Fm(;)p Fq(.)392 1938 y Fm(\()p Fq(:)49 b(If)35 b Fo(x)f Fm(2)f Fo(X)f Fm(n)1069 1913 y Fq(\026)1043 1938 y Fo(A)p Fq(,)37 b(then)f Fo(X)c Fm(n)1618 1913 y Fq(\026)1592 1938 y Fo(A)k Fq(is)f(an)g(op)s(en)h(nhd) g(of)g Fo(x)g Fq(so)f(that,)i(b)m(y)f(h)m(yp)s(othesis,)568 2059 y(\()p Fo(X)30 b Fm(n)814 2034 y Fq(\026)788 2059 y Fo(A)q Fq(\))22 b Fm(\\)g Fo(A)28 b Fm(6)p Fq(=)g Fm(;)p Fq(,)k(whic)m(h)h(is)f(a)g(con)m(tradiction)g(\(i.e.,)g(a)g(false)g (statemen)m(t\).)324 2317 y Fk(Examples)416 2502 y Fq(\(i\))48 b(F)-8 b(or)31 b(an)g(arbitrary)g(in\014nite)g(set)i Fo(X)39 b Fq(with)32 b(the)g(co\014nite)g(top)s(ology)e Fm(C)6 b Fq(,)32 b(the)g(closed)568 2622 y(sets)h(are)g(just)g(the)g (\014nite)f(ones)i(together)e(with)h Fo(X)8 b Fq(.)43 b(So)32 b(for)g(an)m(y)i Fo(A)27 b Fm(\022)i Fo(X)8 b Fq(,)1527 2865 y(\026)1501 2890 y Fo(A)28 b Fq(=)1705 2744 y Ff(\()1814 2829 y Fo(A)99 b Fq(if)31 b Fo(A)i Fq(is)f(\014nite)1814 2950 y Fo(X)91 b Fq(if)31 b Fo(A)i Fq(is)f(in\014nite)568 3158 y(Note)38 b(that)f(an)m(y)i(t)m(w)m(o)f (non-empt)m(y)h(op)s(en)f(subsets)i(of)d Fo(X)46 b Fq(ha)m(v)m(e)39 b(non-empt)m(y)f(in-)568 3279 y(tersection.)389 3480 y(\(ii\))47 b(F)-8 b(or)34 b(an)g(arbitrary)g(uncoun)m(table)h(set)h Fo(X)42 b Fq(with)35 b(the)g(co)s(coun)m(table)g(top)s(ology)e Fm(L)p Fq(,)568 3600 y(the)41 b(closed)g(sets)h(are)f(the)g(coun)m (table)g(ones)h(and)f Fo(X)49 b Fq(itself.)67 b(Note)41 b(that)g(if)f(w)m(e)568 3720 y(let)34 b Fo(X)40 b Fq(=)32 b Fo(R)q Fq(,)k(then)p 1302 3636 196 4 v 35 w([0)p Fo(;)17 b Fq(1])32 b(=)g Fo(R)q Fq(!)51 b(\(In)35 b(the)h(usual)e(Euclidean)h (top)s(ology)-8 b(,)p 3258 3636 V 34 w([0)p Fo(;)17 b Fq(1])32 b(=)568 3841 y([0)p Fo(;)17 b Fq(1].\))362 4042 y(\(iii\))46 b(F)-8 b(or)31 b(the)i(space)h(\()p Fo(X)r(;)17 b Fm(T)1390 4057 y Fj(x)1434 4042 y Fq(\))33 b(de\014ned)h(earlier,)d (if)g Fm(;)d(\032)g Fo(A)g Fm(\022)g Fo(X)8 b Fq(,)32 b(then)1528 4290 y(\026)1502 4315 y Fo(A)c Fq(=)1707 4169 y Ff(\()1815 4254 y Fo(X)340 b Fq(if)31 b Fo(x)d Fm(2)h Fo(A)1815 4375 y(X)h Fm(n)22 b(f)p Fo(x)p Fm(g)83 b Fq(if)31 b Fo(x)d Fm(62)h Fo(A)364 4629 y Fq(\(iv\))49 b(F)-8 b(or)31 b(\()p Fo(X)r(;)17 b Fm(I)7 b Fq(\()p Fo(x)p Fq(\)\))34 b(with)e Fo(A)c Fm(\022)g Fo(X)8 b Fq(,)1652 4877 y(\026)1627 4902 y Fo(A)28 b Fq(=)1831 4756 y Ff(\()1940 4841 y Fo(A)98 b Fq(if)32 b Fo(x)c Fm(62)g Fo(A)1940 4962 y(X)90 b Fq(if)32 b Fo(x)c Fm(2)g Fo(A)1918 5251 y Fq(8)p eop %%Page: 9 10 9 9 bop 392 548 a Fq(\(v\))49 b(F)-8 b(or)31 b(\()p Fo(X)r(;)17 b Fm(E)9 b Fq(\()p Fo(x)p Fq(\)\))33 b(with)f Fm(;)27 b(\032)i Fo(A)e Fm(\022)h Fo(X)8 b Fq(,)1527 805 y(\026)1502 830 y Fo(A)28 b Fq(=)1706 684 y Ff(\()1814 769 y Fo(A)349 b Fq(if)32 b Fo(x)c Fm(2)g Fo(A)1814 890 y(A)23 b Fm([)f(f)p Fo(x)p Fm(g)83 b Fq(if)32 b Fo(x)c Fm(62)g Fo(A)364 1154 y Fq(\(vi\))49 b(In)33 b(\()p Fo(X)r(;)17 b Fm(D)s Fq(\),)32 b(ev)m(ery)i(subset)g(equals)f(its)f(o)m(wn)h(closure.)324 1487 y Fn(1.3)160 b(Con)l(tin)l(uit)l(y)52 b(and)i(Homeomorphism)324 1706 y Fq(The)40 b(cen)m(tral)f(notion)f(of)h(con)m(tin)m(uit)m(y)g(of) g(functions)g(is)f(extended)j(in)e(this)g(section)g(to)324 1826 y(general)f(top)s(ological)d(spaces.)64 b(The)40 b(useful)f(c)m(haracterization)f(of)g(con)m(tin)m(uous)i(func-)324 1947 y(tions)g(in)g(metric)f(spaces)j(as)f(those)g(functions)g(where)h (the)f(in)m(v)m(erse)h(image)d(of)h(ev)m(ery)324 2067 y(op)s(en)33 b(set)g(is)f(op)s(en)h(is)f(used)h(as)g(a)g(de\014nition)e (in)h(the)h(general)f(setting.)324 2187 y(Because)44 b(man)m(y)g(prop)s(erties)f(of)f(spaces)j(are)e(preserv)m(ed)j(b)m(y)e (con)m(tin)m(uous)g(functions,)324 2308 y(spaces)f(related)e(b)m(y)h(a) g(bijection)e(\(one-to-one)h(and)g(on)m(to)h(function\))f(whic)m(h)h (is)f(con-)324 2428 y(tin)m(uous)47 b(in)g(b)s(oth)g(directions)g(will) e(ha)m(v)m(e)k(man)m(y)f(prop)s(erties)f(in)g(common.)87 b(These)324 2549 y(prop)s(erties)27 b(are)g(iden)m(ti\014ed)g(as)h Fp(top)-5 b(olo)g(gic)g(al)29 b(pr)-5 b(op)g(erties)p Fq(.)41 b(Spaces)28 b(so)g(related)f(are)g(called)324 2669 y Fp(home)-5 b(omorphic)p Fq(.)324 2958 y Fl(1.3.1)136 b(Con)l(tin)l(uit)l(y)324 3142 y Fq(The)32 b(primitiv)m(e)d(in)m (tuition)h(of)h(a)g(con)m(tin)m(uous)h(pro)s(cess)h(is)e(that)g(of)g (one)h(in)f(whic)m(h)h(small)324 3263 y(c)m(hanges)41 b(in)f(the)h(input)e(pro)s(duce)i(small,)f('non-catastrophic')g(c)m (hanges)i(in)d(the)i(cor-)324 3383 y(resp)s(onding)d(output.)60 b(This)39 b(idea)e(formalizes)g(easily)g(and)h(naturally)f(for)h (mappings)324 3504 y(from)j(one)h Fp(metric)g Fq(space)h(to)e(another:) 63 b Fo(f)52 b Fq(is)42 b(con)m(tin)m(uous)h(at)e(a)h(p)s(oin)m(t)f Fo(p)h Fq(in)f(suc)m(h)j(a)324 3624 y(setting)35 b(whenev)m(er)j(w)m(e) f(can)f(force)f(the)h(distance)g(b)s(et)m(w)m(een)i Fo(f)11 b Fq(\()p Fo(x)p Fq(\))36 b(and)f Fo(f)11 b Fq(\()p Fo(p)p Fq(\))35 b(to)h(b)s(e)f(as)324 3744 y(small)41 b(as)i(is)f(desired,)47 b(merely)42 b(b)m(y)i(taking)f(the)g(distance)h(b)s(et)m(w)m(een)h Fo(x)e Fq(and)h Fo(p)f Fq(to)f(b)s(e)324 3865 y(small)29 b(enough.)43 b(That)32 b(form)e(of)h(de\014nition)g(is)g(useless)h(in)f (the)h(absence)h(of)e(a)g(prop)s(erly)324 3985 y(de\014ned)41 b('distance')f(function)f(but,)j(fortunately)-8 b(,)41 b(it)e(is)g(equiv)-5 b(alen)m(t)39 b(to)h(the)g(demand)324 4106 y(that)h(the)h(preimage)e(of)h(eac)m(h)i(op)s(en)f(subset)h(of)e (the)h(target)f(metric)g(space)h(shall)e(b)s(e)324 4226 y(op)s(en)d(in)g(the)g(domain.)56 b(Th)m(us)39 b(expressed,)i(the)c (idea)g(is)g(immediately)d(transferrable)324 4346 y(to)e(general)g(top) s(ology:)324 4695 y Fk(De\014nition)k(1.8)49 b Fp(L)-5 b(et)40 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))40 b Fp(and)f Fq(\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))40 b Fp(b)-5 b(e)40 b(top)-5 b(olo)g(gic)g(al)39 b(sp)-5 b(ac)g(es;)41 b(a)f(mapping)f Fo(f)48 b Fq(:)324 4815 y Fo(X)35 b Fm(!)28 b Fo(Y)52 b Fp(is)32 b(c)-5 b(al)5 b(le)-5 b(d)30 b Fk(con)m(tin)m (uous)h Fp(i\013)h Fo(f)1780 4779 y Fg(\000)p Fh(1)1874 4815 y Fq(\()p Fo(U)10 b Fq(\))28 b Fm(2)g(T)57 b Fp(for)31 b(e)-5 b(ach)31 b Fo(U)38 b Fm(2)28 b(S)40 b Fp(i.e.)j(the)32 b(inverse)324 4936 y(image)i(of)g(any)h(op)-5 b(en)34 b(subset)h(of)g Fo(Y)56 b Fp(is)34 b(op)-5 b(en)35 b(in)f Fo(X)8 b Fp(.)1918 5251 y Fq(9)p eop %%Page: 10 11 10 10 bop 324 548 a Fp(Examples)416 723 y Fq(\(i\))48 b(If)35 b(\()p Fo(X)r(;)17 b Fm(D)s Fq(\))35 b(is)g(discrete)h(and)f (\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))36 b(is)f(an)g(arbitrary)g(top)s (ological)d(space,)37 b(then)568 844 y Fp(any)32 b Fq(function)g Fo(f)39 b Fq(:)27 b Fo(X)36 b Fm(!)27 b Fo(Y)54 b Fq(is)32 b(con)m(tin)m(uous!)568 1001 y(Again,)f(if)g(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))32 b(is)f(an)h(arbitrary)f(top)s(ological)d(space)33 b(and)g(\()p Fo(Y)5 b(;)17 b Fm(T)3071 1016 y Fh(0)3110 1001 y Fq(\))32 b(is)f(trivial,)568 1121 y Fp(any)h Fq(mapping)f Fo(g)g Fq(:)d Fo(X)35 b Fm(!)28 b Fo(Y)53 b Fq(is)33 b(con)m(tin)m(uous.)389 1315 y(\(ii\))47 b(If)36 b(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\),)37 b(\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))37 b(are)g(arbitrary)e(top)s(ological)e(spaces)38 b(and)f Fo(f)45 b Fq(:)35 b Fo(X)42 b Fm(!)34 b Fo(Y)58 b Fq(is)36 b(a)568 1435 y(constan)m(t)d(map,)f(then)h Fo(f)43 b Fq(is)33 b(con)m(tin)m(uous.)362 1630 y(\(iii\))46 b(Let)28 b Fo(X)36 b Fq(b)s(e)29 b(an)g(arbitrary)e(set)i(ha)m(ving)f (more)g(than)h(t)m(w)m(o)g(elemen)m(ts,)g(with)g Fo(x)f Fm(2)g Fo(X)8 b Fq(.)568 1750 y(Let)32 b Fm(T)53 b Fq(=)28 b Fm(I)7 b Fq(\()p Fo(x)p Fq(\),)33 b Fm(S)i Fq(=)28 b Fm(T)1458 1765 y Fj(x)1534 1750 y Fq(in)j(the)h(de\014nition)f(of)h (con)m(tin)m(uit)m(y;)g(then)g(the)h(iden)m(tit)m(y)568 1870 y(map)j Fo(id)873 1885 y Fj(X)975 1870 y Fq(:)f Fo(X)43 b Fm(!)35 b Fo(X)45 b Fq(is)36 b(con)m(tin)m(uous.)58 b(Ho)m(w)m(ev)m(er,)40 b(if)c(w)m(e)i(in)m(terc)m(hange)f Fm(T)63 b Fq(with)568 1991 y Fm(S)40 b Fq(so)33 b(that)g Fm(T)54 b Fq(=)28 b Fm(T)1266 2006 y Fj(x)1343 1991 y Fq(and)33 b Fm(S)j Fq(=)28 b Fm(I)7 b Fq(\()p Fo(x)p Fq(\),)34 b(then)g Fo(id)2293 2006 y Fj(X)2388 1991 y Fq(:)29 b Fo(X)36 b Fm(!)28 b Fo(X)40 b Fq(is)33 b Fp(not)g Fq(con)m(tin)m(uous!)568 2111 y(Note)28 b(that)g Fo(id)1090 2126 y Fj(X)1185 2111 y Fq(:)f(\()p Fo(X)r(;)17 b Fm(T)1458 2126 y Fh(1)1498 2111 y Fq(\))28 b Fm(!)f Fq(\()p Fo(X)r(;)17 b Fm(T)1910 2126 y Fh(2)1950 2111 y Fq(\))27 b(is)h(con)m(tin)m(uous)h (if)d(and)j(only)e(if)g Fm(T)3211 2126 y Fh(1)3278 2111 y Fq(is)h(\014ner)568 2231 y(than)k Fm(T)849 2246 y Fh(2)889 2231 y Fq(.)324 2422 y Fk(Theorem)37 b(1.2)49 b Fp(If)41 b Fq(\()p Fo(X)1219 2437 y Fh(1)1259 2422 y Fo(;)17 b Fm(T)1357 2437 y Fh(1)1396 2422 y Fq(\))p Fp(,)44 b Fq(\()p Fo(X)1627 2437 y Fh(2)1666 2422 y Fo(;)17 b Fm(T)1764 2437 y Fh(2)1803 2422 y Fq(\))42 b Fp(and)g Fq(\()p Fo(X)2199 2437 y Fh(3)2238 2422 y Fo(;)17 b Fm(T)2336 2437 y Fh(3)2376 2422 y Fq(\))41 b Fp(ar)-5 b(e)42 b(top)-5 b(olo)g(gic)g(al)41 b(sp)-5 b(ac)g(es)41 b(and)324 2543 y Fo(h)g Fq(:)h Fo(X)571 2558 y Fh(1)652 2543 y Fm(!)f Fo(X)874 2558 y Fh(2)955 2543 y Fp(and)h Fo(g)j Fq(:)c Fo(X)1393 2558 y Fh(2)1474 2543 y Fm(!)g Fo(X)1696 2558 y Fh(3)1778 2543 y Fp(ar)-5 b(e)42 b(c)-5 b(ontinuous,)44 b(then)e Fo(g)31 b Fm(\016)c Fo(h)42 b Fq(:)f Fo(X)3107 2558 y Fh(1)3188 2543 y Fm(!)g Fo(X)3410 2558 y Fh(3)3492 2543 y Fp(is)324 2663 y(c)-5 b(ontinuous.)324 2854 y Fq(Pro)s(of)p 324 2867 235 4 v 32 w(Immediate.)324 3094 y(There)37 b(are)f(sev)m(eral)g(di\013eren)m (t)g(w)m(a)m(ys)i(to)d('recognise')i(con)m(tin)m(uit)m(y)f(for)f(a)h (mapping)e(b)s(e-)324 3215 y(t)m(w)m(een)26 b(top)s(ological)21 b(spaces,)28 b(of)c(whic)m(h)h(the)g(next)h(theorem)e(indicates)g(t)m (w)m(o)h(of)f(the)h(most)324 3335 y(useful)32 b(apart)h(from)e(the)i (de\014nition)e(itself:)324 3646 y Fk(Theorem)37 b(1.3)49 b Fp(L)-5 b(et)47 b Fo(f)58 b Fp(b)-5 b(e)47 b(a)g(mapping)f(fr)-5 b(om)47 b(a)g(top)-5 b(olo)g(gic)g(al)46 b(sp)-5 b(ac)g(e)47 b Fq(\()p Fo(X)3121 3661 y Fh(1)3160 3646 y Fo(;)17 b Fm(T)3258 3661 y Fh(1)3297 3646 y Fq(\))47 b Fp(to)h(a)324 3767 y(top)-5 b(olo)g(gic)g(al)34 b(sp)-5 b(ac)g(e)34 b Fq(\()p Fo(X)1168 3782 y Fh(2)1207 3767 y Fo(;)17 b Fm(T)1305 3782 y Fh(2)1344 3767 y Fq(\))p Fp(.)45 b(The)34 b(fol)5 b(lowing)34 b(statements)g(ar)-5 b(e)35 b(e)-5 b(quivalent:)409 4062 y(\(i\))49 b Fo(f)c Fp(is)35 b(c)-5 b(ontinuous,)379 4256 y(\(ii\))49 b(the)35 b(pr)-5 b(eimage)33 b(under)i Fo(f)46 b Fp(of)34 b(e)-5 b(ach)34 b(close)-5 b(d)34 b(subset)h(of)g Fo(X)2618 4271 y Fh(2)2692 4256 y Fp(is)g(close)-5 b(d)34 b(in)g Fo(X)3281 4271 y Fh(1)3320 4256 y Fp(,)350 4450 y(\(iii\))48 b(for)34 b(every)h(subset)g Fo(A)g Fp(of)g Fo(X)1574 4465 y Fh(1)1613 4450 y Fp(,)g Fo(f)11 b Fq(\()1800 4425 y(\026)1775 4450 y Fo(A)p Fq(\))27 b Fm(\022)p 2018 4366 208 4 v 28 w Fo(f)11 b Fq(\()p Fo(A)p Fq(\))p Fp(.)324 4641 y Fq(Pro)s(of)p 324 4654 235 4 v 29 w(It)30 b(is)f(easy)i(to)e(see)i(that)e(\(i\))g(implies)e (\(ii\).)41 b(Assuming)29 b(that)g(\(ii\))f(holds,)i(apply)g(it)324 4761 y(to)f(the)g(closed)h(set)p 1038 4677 208 4 v 30 w Fo(f)11 b Fq(\()p Fo(A)p Fq(\))29 b(and)g(\(iii\))e(readily)h(follo)m (ws.)41 b(No)m(w)30 b(if)e(\(iii\))e(is)j(assumed)h(and)g Fo(G)324 4882 y Fq(is)h(a)g(giv)m(en)g(op)s(en)h(subset)h(of)e Fo(X)1478 4897 y Fh(2)1517 4882 y Fq(,)h(use)g(\(iii\))c(on)k(the)g (set)g Fo(A)27 b Fq(=)h Fo(X)2669 4897 y Fh(1)2728 4882 y Fm(n)20 b Fo(f)2857 4846 y Fg(\000)p Fh(1)2951 4882 y Fq(\()p Fo(G)p Fq(\))31 b(and)g(v)m(erify)324 5002 y(that)h(it)g(follo)m(ws)f(that)h Fo(f)1223 4966 y Fg(\000)p Fh(1)1317 5002 y Fq(\()p Fo(G)p Fq(\))h(m)m(ust)f(b)s(e)h(op)s(en.)1894 5251 y(10)p eop %%Page: 11 12 11 11 bop 324 548 a Fl(1.3.2)136 b(Homeomorphism)324 733 y Fk(De\014nition)36 b(1.9)49 b Fp(L)-5 b(et)38 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))p Fp(,)39 b Fq(\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))38 b Fp(b)-5 b(e)38 b(top)-5 b(olo)g(gic)g(al)37 b(sp)-5 b(ac)g(es)38 b(and)f(let)h Fo(h)c Fq(:)g Fo(X)42 b Fm(!)33 b Fo(Y)324 853 y Fp(b)-5 b(e)39 b(bije)-5 b(ctive.)57 b(Then)39 b Fo(h)g Fp(is)h(a)f Fk(homeomorphism)f Fp(i\013)h Fo(h)g Fp(is)g(c)-5 b(ontinuous)39 b Fq(and)h Fo(h)3358 817 y Fg(\000)p Fh(1)3492 853 y Fp(is)324 973 y(c)-5 b(ontinuous.)47 b(If)35 b(such)g(a)h(map)f (exists,)g Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))36 b Fp(and)f Fq(\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))36 b Fp(ar)-5 b(e)35 b(c)-5 b(al)5 b(le)-5 b(d)35 b Fk(homeomor-)324 1094 y(phic)p Fp(.)324 1318 y Fq(Suc)m(h)f(a)e(map)g(has)h(the)g(prop)s (ert)m(y)g(that)1512 1535 y Fo(G)28 b Fm(2)g(T)53 b(,)27 b Fo(f)11 b Fq(\()p Fo(G)p Fq(\))27 b Fm(2)i(S)7 b Fo(:)324 1752 y Fq(It)35 b(follo)m(ws)g(that)g(an)m(y)h(statemen)m(t)g(ab)s(out) f(a)h(top)s(ological)c(space)k(whic)m(h)g(is)f(ultimately)324 1872 y(expressible)e(solely)e(in)g(terms)h(of)f(the)i(op)s(en)f(sets)h (\(together)f(with)g(set-theoretic)g(rela-)324 1993 y(tions)g(and)g(op) s(erations\))g(will)e(b)s(e)j(true)g(for)f(b)s(oth)g(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))33 b(and)f(\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))33 b(if)f(it)f(is)h(true)h(for)324 2113 y(either.)42 b(In)28 b(other)g(w)m(ords,)i(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))i(and)g(\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))28 b(are)h(indistinguishable)c(as)j(top)s(ological)324 2233 y(spaces.)89 b(The)48 b(reader)g(who)f(has)h(had)f(abstract)g (algebra)g(will)d(note)k(that)f(homeo-)324 2354 y(morphism)31 b(is)h(the)h(analogy)f(in)g(the)h(setting)f(of)h(top)s(ological)c (spaces)34 b(and)f(con)m(tin)m(uous)324 2474 y(functions)27 b(to)f(the)h(notion)f(of)h(isomorphism)d(in)i(the)i(setting)e(of)h (groups)g(\(or)f(rings\))g(and)324 2594 y(homomorphisms,)31 b(and)j(to)f(that)h(of)f(linear)f(isomorphism)f(in)i(the)h(con)m(text)g (of)g(v)m(ector)324 2715 y(spaces)g(and)f(linear)e(maps.)324 2835 y Fp(Example)324 2956 y Fq(F)-8 b(or)33 b(ev)m(ery)i(space)g(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\),)34 b(the)g(iden)m(tit)m(y)g(mapping)e Fo(id)2380 2971 y Fj(X)2477 2956 y Fq(:)d Fo(X)38 b Fm(!)29 b Fo(X)41 b Fq(is)33 b(a)h(homeomor-)324 3076 y(phism.)324 3196 y(A)26 b(prop)s(ert)m(y)g(of)g(top)s(ological)c(spaces)27 b(whic)m(h)g(when)g(p)s(ossessed)h(b)m(y)f(a)f(space)h(is)e(also)g(p)s (os-)324 3317 y(sessed)k(b)m(y)f(ev)m(ery)h(space)g(homeomorphic)c(to)i (it)f(is)h(called)f(a)h Fk(top)s(ological)i(in)m(v)-6 b(arian)m(t)p Fq(.)324 3437 y(W)e(e)33 b(shall)e(meet)h(some)h (examples)f(of)g(suc)m(h)i(prop)s(erties)f(later.)324 3557 y(One)g(can)f(readily)g(v)m(erify)h(that)f(if)f Fo(f)43 b Fq(is)32 b(a)h(homeomorphism,)d(then)j(the)g(in)m(v)m(erse)h (map-)324 3678 y(ping)40 b Fo(f)608 3642 y Fg(\000)p Fh(1)743 3678 y Fq(is)h(also)f(a)h(homeomorphism)e(and)j(that)f(the)g (comp)s(osition)e Fo(g)31 b Fm(\016)d Fo(f)52 b Fq(of)41 b(t)m(w)m(o)324 3798 y(homeomorphisms)28 b Fo(f)41 b Fq(and)31 b Fo(g)i Fq(is)d(again)f(a)i(homeomorphism.)40 b(Th)m(us,)32 b(the)f(relation)e(`)p Fo(X)324 3919 y Fq(and)j Fo(Y)54 b Fq(are)33 b(homeomorphic')e(is)h(an)g(equiv)-5 b(alence)33 b(relation.)324 4039 y(In)26 b(general,)g(it)f(ma)m(y)g(b)s (e)h(quite)g(di\016cult)f(to)g(demonstrate)h(that)f(t)m(w)m(o)i(spaces) g(are)f(home-)324 4159 y(omorphic)33 b(\(unless)i(a)f(homeomorphism)e (is)i(ob)m(vious)g(or)g(can)h(easily)e(b)s(e)i(disco)m(v)m(ered\).)324 4280 y(F)-8 b(or)45 b(example,)50 b(to)c(v)m(erify)g(that)h(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\))46 b(is)g(homeomorphic)f(to)h(\(0)p Fo(;)17 b Fq(1\))45 b(with)h(its)g(in-)324 4400 y(duced)j(metric)e(top) s(ology)-8 b(,)50 b(it)d(is)g(necessary)k(to)c(demonstrate,)52 b(for)c(instance,)k(that)324 4521 y Fo(h)28 b Fq(:)f(\(0)p Fo(;)17 b Fq(1\))27 b Fm(!)g Fo(R)34 b Fq(where)g Fo(h)p Fq(\()p Fo(x)p Fq(\))28 b(=)1582 4481 y Fh(2)p Fj(x)p Fg(\000)p Fh(1)p 1553 4497 225 4 v 1553 4555 a Fj(x)p Fh(\()p Fj(x)p Fg(\000)p Fh(1\))1820 4521 y Fq(is)k(a)g(homeomorphism.) 324 4641 y(It)44 b(is)g(often)h(easier)f(to)g(sho)m(w)i(that)e(t)m(w)m (o)h(spaces)i(are)d Fp(not)g Fq(homeomorphic:)66 b(simply)324 4761 y(exhibit)32 b(an)g(in)m(v)-5 b(arian)m(t)31 b(whic)m(h)i(is)g(p)s (ossessed)i(b)m(y)e(one)g(space)h(and)e(not)h(the)g(other.)324 4882 y Fp(Example)324 5002 y Fq(The)24 b(spaces)i(\()p Fo(X)r(;)17 b Fm(I)7 b Fq(\()p Fo(x)p Fq(\)\))24 b(and)g(\()p Fo(X)r(;)17 b Fm(E)9 b Fq(\()p Fo(x)p Fq(\)\))23 b(are)h Fp(not)f Fq(homeomorphic)f(since,)k(for)d(example,)1894 5251 y(11)p eop %%Page: 12 13 12 12 bop 324 548 a Fq(\()p Fo(X)r(;)17 b Fm(I)7 b Fq(\()p Fo(x)p Fq(\)\))23 b(has)g(the)g(top)s(ological)c(in)m(v)-5 b(arian)m(t)21 b(`eac)m(h)i(nhd)g(is)f(op)s(en')h(while)f(\()p Fo(X)r(;)17 b Fm(E)9 b Fq(\()p Fo(x)p Fq(\)\))22 b(do)s(es)324 668 y(not.)324 994 y Fn(1.4)160 b(Additional)55 b(Observ)-9 b(ations)324 1213 y Fk(De\014nition)36 b(1.10)49 b Fp(A)c Fq(sequence)j(\()p Fo(x)1722 1228 y Fj(n)1770 1213 y Fq(\))d Fp(in)f(a)h(top)-5 b(olo)g(gic)g(al)44 b(sp)-5 b(ac)g(e)45 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))45 b Fp(is)g(said)f(to)324 1334 y(c)-5 b(onver)g(ge)37 b(to)h(a)g(p)-5 b(oint)38 b Fo(x)c Fm(2)g Fo(X)46 b Fp(i\013)38 b Fq(\()p Fo(x)1717 1349 y Fj(n)1765 1334 y Fq(\))g Fp(eventual)5 b(ly)38 b(b)-5 b(elongs)37 b(to)h(every)g(nhd)g(of)g Fo(x)g Fp(i.e.)324 1454 y(i\013)30 b(for)h(every)g(nhd)f Fo(U)42 b Fp(of)31 b Fo(x)p Fp(,)h(ther)-5 b(e)30 b(exists)h Fo(n)1930 1469 y Fh(0)1997 1454 y Fm(2)d Fo(N)42 b Fp(such)31 b(that)g Fo(x)2679 1469 y Fj(n)2754 1454 y Fm(2)d Fo(U)42 b Fp(for)30 b(al)5 b(l)31 b Fo(n)d Fm(\025)g Fo(n)3492 1469 y Fh(0)3532 1454 y Fp(.)324 1630 y Fk(Caution)324 1751 y Fq(W)-8 b(e)41 b(learn)m(t)g(that,)i(for)e(metric)f(spaces,)45 b(sequen)m(tial)c(con)m(v)m(ergence)j(w)m(as)e(adequate)g(to)324 1871 y(describ)s(e)37 b(the)f(top)s(ology)f(of)h(suc)m(h)i(spaces)f (\(in)f(the)h(sense)h(that)e(the)g(basic)g(primitiv)m(es)324 1991 y(of)g(`op)s(en)h(set',)h(`neigh)m(b)s(ourho)s(o)s(d',)f (`closure')g(etc.)56 b(could)36 b(b)s(e)h(fully)e(c)m(haracterised)i (in)324 2112 y(terms)k(of)g(sequen)m(tial)g(con)m(v)m(ergence\).)73 b(Ho)m(w)m(ev)m(er,)46 b(for)41 b(general)f(top)s(ological)e(spaces,) 324 2232 y(sequen)m(tial)33 b(con)m(v)m(ergence)i(fails.)42 b(W)-8 b(e)32 b(illustrate:)416 2397 y(\(i\))48 b(Limits)34 b(are)i Fp(not)g Fq(alw)m(a)m(ys)h(unique.)55 b(F)-8 b(or)36 b(example,)h(in)e(\()p Fo(X)r(;)17 b Fm(T)2827 2412 y Fh(0)2867 2397 y Fq(\),)37 b Fp(e)-5 b(ach)36 b Fq(sequence)568 2517 y(\()p Fo(x)661 2532 y Fj(n)708 2517 y Fq(\))c(con)m(v)m(erges)j(to)d Fp(every)h Fo(x)28 b Fm(2)g Fo(X)8 b Fq(.)389 2707 y(\(ii\))47 b(In)53 b Fo(R)h Fq(with)e(the)h(co)s(coun)m(table)g(top)s(ology)e Fm(L)p Fq(,)58 b([0)p Fo(;)17 b Fq(1])52 b(is)h Fp(not)f Fq(closed)h(and)g(so)568 2828 y Fo(G)46 b Fq(=)h(\()p Fm(\0001)p Fo(;)17 b Fq(0\))29 b Fm([)i Fq(\(1)p Fo(;)17 b Fm(1)p Fq(\))42 b(is)i(not)f(op)s(en)h(|)g(y)m(et)g(if)f Fo(x)2609 2843 y Fj(n)2703 2828 y Fm(!)k Fo(x)d Fq(where)h Fo(x)i Fm(2)h Fo(G)p Fq(,)568 2948 y(then)34 b(Assignmen)m(t)g(1)f(sho) m(ws)j(that)d Fo(x)1950 2963 y Fj(n)2027 2948 y Fm(2)d Fo(G)k Fq(for)f(all)e(su\016cien)m(tly)k(large)d Fo(n)p Fq(.)47 b(F)-8 b(ur-)568 3086 y(ther,)37 b(2)d Fm(2)p 988 3001 196 4 v 34 w Fq([0)p Fo(;)17 b Fq(1])1183 3021 y Fg(L)1236 3086 y Fq(,)37 b(y)m(et)g(no)f(sequence)j(in)c([0)p Fo(;)17 b Fq(1])36 b(can)g(approac)m(h)h(2.)53 b(So)36 b(another)568 3207 y(c)m(haracterisation)31 b(fails)g(to)h(carry)h(o)m (v)m(er)h(from)d(metric)h(space)i(theory)-8 b(.)568 3362 y(Finally)g(,)39 b(ev)m(ery)j Fm(L)p Fq(-con)m(v)m(ergen)m(t)f (sequence)i(of)c(p)s(oin)m(ts)h(in)f([0)p Fo(;)17 b Fq(1])40 b(m)m(ust)g(ha)m(v)m(e)h(its)568 3482 y(limit)29 b Fp(in)j Fq([0)p Fo(;)17 b Fq(1])32 b(|)g(but)h([0)p Fo(;)17 b Fq(1])32 b(is)g Fp(not)g Fq(closed)h(\(in)f Fm(L)p Fq(\)!)324 3647 y(Hence,)38 b(to)d(discuss)i(top)s(ological)32 b(con)m(v)m (ergence)39 b(thoroughly)-8 b(,)35 b(w)m(e)i(need)g(to)e(dev)m(elop)i (a)324 3767 y(new)i(basic)g(set-theoretic)f(to)s(ol)f(whic)m(h)i (generalises)g(the)g(notion)e(of)h(sequence.)64 b(It)39 b(is)324 3887 y(called)31 b(a)h Fk(net)h Fq(|)f(w)m(e)i(shall)d(return) i(to)f(this)g(later.)324 4063 y Fk(De\014nition)k(1.11)49 b Fp(A)43 b(top)-5 b(olo)g(gic)g(al)42 b(sp)-5 b(ac)g(e)41 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))43 b Fp(is)f(c)-5 b(al)5 b(le)-5 b(d)42 b Fk(metrizable)e Fp(i\013)j(ther)-5 b(e)324 4184 y(exists)33 b(a)h(metric)g Fo(d)f Fp(on)h Fo(X)41 b Fp(such)34 b(that)g(the)g(top)-5 b(olo)g(gy)34 b Fm(T)2332 4199 y Fj(d)2407 4184 y Fp(induc)-5 b(e)g(d)33 b(by)h Fo(d)f Fp(c)-5 b(oincides)33 b(with)324 4304 y(the)i(original)f (top)-5 b(olo)g(gy)34 b Fm(T)61 b Fp(on)34 b Fo(X)8 b Fp(.)324 4480 y Fq(The)33 b(in)m(v)m(estigations)f(ab)s(o)m(v)m(e)h (sho)m(w)h(that)e(\()p Fo(X)r(;)17 b Fm(T)2083 4495 y Fh(0)2123 4480 y Fq(\))32 b(and)g(\()p Fo(R)q(;)17 b Fm(L)p Fq(\))32 b(are)h(examples)f(of)g(non-)324 4601 y(metrizable)k(spaces.)60 b(Ho)m(w)m(ev)m(er,)41 b(the)d(discrete)g (space)h(\()p Fo(X)r(;)17 b Fm(D)s Fq(\))37 b Fp(is)45 b Fq(metrizable,)37 b(b)s(eing)324 4721 y(induced)c(b)m(y)g(the)g (discrete)h(metric)1429 4952 y Fo(d)p Fq(\()p Fo(x;)17 b(y)t Fq(\))27 b(=)1837 4806 y Ff(\()1946 4891 y Fq(1)82 b(if)32 b Fo(x)c Fm(6)p Fq(=)g Fo(y)1946 5012 y Fq(0)82 b(if)32 b Fo(x)c Fq(=)g Fo(y)1894 5251 y Fq(12)p eop %%Page: 13 14 13 13 bop 324 1212 a Fr(Chapter)78 b(2)324 1627 y(T)-19 b(op)6 b(ological)77 b(Prop)6 b(erties)324 2080 y Fq(W)-8 b(e)38 b(explained)f(in)f(the)i(previous)g(c)m(hapter)g(what)g(a)f(top) s(ological)d(prop)s(ert)m(y)k(\(homeo-)324 2200 y(morphic)32 b(in)m(v)-5 b(arian)m(t\))32 b(is)g(but)i(ga)m(v)m(e)g(few)g(go)s(o)s (d)e(examples.)45 b(W)-8 b(e)34 b(no)m(w)g(explore)f(some)g(of)324 2320 y(the)g(most)f(imp)s(ortan)m(t)e(ones.)45 b(Recurring)32 b(themes)h(will)d(b)s(e:)469 2505 y Fm(\017)49 b Fq(When)33 b(do)g(subspaces)i(inherit)c(the)i(prop)s(ert)m(y?)469 2702 y Fm(\017)49 b Fq(Ho)m(w)33 b(do)f(con)m(tin)m(uous)i(maps)e (relate)g(to)g(the)h(prop)s(ert)m(y?)469 2900 y Fm(\017)49 b Fq(Do)s(es)32 b(the)h(prop)s(ert)m(y)g(b)s(eha)m(v)m(e)h(sp)s (ecially)e(in)f(metric)h(spaces?)324 3229 y Fn(2.1)160 b(Compactness)324 3448 y Fq(W)-8 b(e)30 b(all)e(recall)g(the)j(imp)s (ortan)m(t)d(and)i(useful)g(theorem)f(from)g(calculus,)h(that)g (functions)324 3569 y(whic)m(h)k(are)g(con)m(tin)m(uous)g(on)f(a)h (closed)f(and)h(b)s(ounded)g(in)m(terv)-5 b(al)33 b(tak)m(e)h(on)g(a)f (maxim)m(um)324 3689 y(and)41 b(minim)m(um)c(v)-5 b(alue)41 b(on)g(that)f(in)m(terv)-5 b(al.)68 b(The)42 b(classic)e(theorem)h(of)f (Heine-Borel-)324 3809 y(Leb)s(esgue)d(asserts)h(that)d(ev)m(ery)j(co)m (v)m(ering)f(of)f(suc)m(h)h(an)f(in)m(terv)-5 b(al)35 b(b)m(y)i(op)s(en)f(sets)h(has)g(a)324 3930 y(\014nite)i(sub)s(co)m(v)m (er.)67 b(In)39 b(this)h(section,)h(w)m(e)g(use)f(this)f(feature)h(of)f (closed)h(and)g(b)s(ounded)324 4050 y(subsets)e(to)e(de\014ne)h(the)g (corresp)s(onding)f(notion,)g Fp(c)-5 b(omp)g(actness)p Fq(,)36 b(in)f(a)h(general)g(top)s(o-)324 4171 y(logical)41 b(space.)82 b(In)45 b(addition,)h(w)m(e)g(consider)f(imp)s(ortan)m(t)e (v)-5 b(arian)m(ts)44 b(of)g(this)h(notion:)324 4291 y(sequen)m(tial)33 b(compactness)g(and)g(lo)s(cal)d(compactness.)324 4577 y Fl(2.1.1)136 b(Compactness)45 b(De\014ned)324 4761 y Fq(Giv)m(en)26 b(a)f(set)i Fo(X)33 b Fq(with)26 b Fo(A)i Fm(\022)g Fo(X)8 b Fq(,)27 b(a)e Fp(c)-5 b(over)36 b Fq(for)26 b Fo(A)g Fq(is)f(a)h(family)d(of)i(subsets)j Fm(U)38 b Fq(=)28 b Fm(f)p Fo(U)3284 4776 y Fj(i)3340 4761 y Fq(:)p Fm(2)g Fo(I)8 b Fm(g)324 4882 y Fq(of)24 b Fo(X)32 b Fq(suc)m(h)26 b(that)e Fo(A)k Fm(\022)g([)1227 4897 y Fj(i)p Fg(2)p Fj(I)1338 4882 y Fo(U)1404 4897 y Fj(i)1433 4882 y Fq(.)40 b(A)25 b Fp(sub)-5 b(c)g(over)34 b Fq(of)24 b(a)g(giv)m(en)h(co)m(v)m(er)g Fm(U)35 b Fq(for)24 b Fo(A)g Fq(is)g(a)g(subfamily)324 5002 y Fm(V)36 b(\032)28 b(U)43 b Fq(whic)m(h)33 b(still)d(forms)i(a)g(co)m(v)m(er)i(for)e Fo(A)p Fq(.)1894 5251 y(13)p eop %%Page: 14 15 14 14 bop 324 548 a Fq(If)30 b Fo(A)g Fq(is)f(a)g(subspace)j(of)e(a)f (space)i(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\),)k Fm(U)40 b Fq(is)30 b(an)f Fp(op)-5 b(en)32 b(c)-5 b(over)40 b Fq(for)29 b Fo(A)h Fq(i\013)f Fm(U)40 b Fq(is)30 b(a)f(co)m(v)m(er)324 668 y(for)j Fo(A)g Fq(suc)m(h)i(that)f(eac)m(h)g(mem)m(b)s(er)f(of)g Fm(U)43 b Fq(is)32 b(op)s(en)h(in)f Fo(X)8 b Fq(.)324 789 y(The)36 b(classic)f(theorem)g(of)g(Heine-Borel-Leb)s(esgue)g (asserts)i(that,)f(in)f Fo(R)q Fq(,)h(ev)m(ery)h(op)s(en)324 909 y(co)m(v)m(er)42 b(of)f(a)g(closed)g(b)s(ounded)h(subset)h(has)f(a) f(\014nite)g(sub)s(co)m(v)m(er.)71 b(This)41 b(theorem)g(has)324 1029 y(extraordinarily)26 b(profound)i(consequences)k(and)c(lik)m(e)g (most)g(go)s(o)s(d)f(theorems,)i(its)f(con-)324 1150 y(clusion)j(has)i(b)s(ecome)g(a)f(de\014nition.)324 1378 y Fk(De\014nition)k(2.1)49 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))33 b Fp(is)f(said)g(to)h(b)-5 b(e)32 b Fk(compact)g Fp(i\013)g(every)g(op)-5 b(en)32 b(c)-5 b(over)32 b(of)g Fo(X)40 b Fp(has)324 1499 y(a)34 b(\014nite)h(sub)-5 b(c)g(over.)324 1727 y Fk(Theorem)37 b(2.1)g(\(Alexander's)h(Subbase)h (Theorem\))48 b Fp(L)-5 b(et)53 b Fm(S)59 b Fp(b)-5 b(e)52 b(any)g(subb)-5 b(ase)324 1847 y(for)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))p Fp(.)45 b(If)34 b(every)h(op)-5 b(en)34 b(c)-5 b(over)34 b(of)h Fo(X)43 b Fp(by)35 b(memb)-5 b(ers)33 b(of)i Fm(S)43 b Fp(has)34 b(a)h(\014nite)f(sub)-5 b(c)g(over,)324 1968 y(then)34 b Fo(X)43 b Fp(is)35 b(c)-5 b(omp)g(act.)324 2196 y Fq(The)38 b(pro)s(of)e(of)h(this)g(deep)h (result)f(is)f(an)h(application)e(of)i(Zorn's)g(lemma,)f(and)h(is)g (not)324 2316 y(an)32 b(exercise)i(for)e(the)h(fain)m(t-hearted!)324 2696 y Fk(Examples)416 2881 y Fq(\(i\))48 b(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\))35 b(is)f Fp(not)44 b Fq(compact,)35 b(for)f(consider)h Fm(U)42 b Fq(=)31 b Fm(f)p Fq(\()p Fm(\000)p Fo(n;)17 b(n)p Fq(\))32 b(:)g Fo(n)f Fm(2)h Fo(N)10 b Fm(g)p Fq(.)51 b(Similarly)-8 b(,)568 3002 y(\()p Fo(C)r(;)17 b Fm(T)776 3017 y Fj(usual)953 3002 y Fq(\))32 b(is)g(not)h(compact.)389 3325 y(\(ii\))47 b(\(0)p Fo(;)17 b Fq(1\))31 b(is)h Fp(not)42 b Fq(compact,)33 b(for)f(consider)h Fm(U)38 b Fq(=)27 b Fm(f)p Fq(\()2346 3286 y Fh(1)p 2342 3302 43 4 v 2342 3360 a Fj(n)2395 3325 y Fo(;)17 b Fq(1\))27 b(:)g Fo(n)h Fm(\025)g Fq(2)p Fm(g)p Fq(.)362 3649 y(\(iii\))46 b(\()p Fo(X)r(;)17 b Fm(C)6 b Fq(\))32 b Fp(is)41 b Fq(compact,)32 b(for)g(an)m(y)h Fo(X)8 b Fq(.)364 3973 y(\(iv\))49 b(Giv)m(en)29 b Fo(x)f Fm(2)g Fo(X)8 b Fq(,)30 b(\()p Fo(X)r(;)17 b Fm(E)9 b Fq(\()p Fo(x)p Fq(\)\))29 b(is)g(compact;)h(\()p Fo(X)r(;)17 b Fm(I)7 b Fq(\()p Fo(x)p Fq(\)\))30 b(is)f Fp(not)39 b Fq(compact)29 b(unless)h Fo(X)568 4093 y Fq(is)i(\014nite.)392 4417 y(\(v\))49 b Fm(T)58 b Fq(\014nite)32 b(for)g Fp(any)41 b Fo(X)36 b Fm(\))27 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))32 b(compact.)364 4741 y(\(vi\))49 b Fo(X)40 b Fq(\014nite,)32 b Fm(T)58 b Fq(an)m(y)34 b(top)s(ology)d(for)h Fo(X)j Fm(\))28 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))33 b(compact.)1894 5251 y(14)p eop %%Page: 15 16 15 15 bop 337 548 a Fq(\(vii\))48 b Fo(X)40 b Fq(in\014nite)32 b Fm(\))27 b Fq(\()p Fo(X)r(;)17 b Fm(D)s Fq(\))32 b(not)g(compact.)310 860 y(\(viii\))47 b(Giv)m(en)39 b(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\),)41 b(if)d(\()p Fo(x)1396 875 y Fj(n)1443 860 y Fq(\))h(is)g(a)g(sequence)i(in)e Fo(X)47 b Fq(con)m(v)m(ergen)m (t)41 b(to)e Fo(x)p Fq(,)i(then)f Fm(f)p Fo(x)3449 875 y Fj(n)3535 860 y Fq(:)568 981 y Fo(n)28 b Fm(2)g Fo(N)10 b Fm(g)22 b([)h(f)p Fo(x)p Fm(g)32 b Fq(is)g(compact.)324 1264 y Fl(2.1.2)136 b(Compactness)45 b(for)h(Subspaces)324 1448 y Fq(W)-8 b(e)32 b(call)f(a)g(subset)j Fo(A)e Fq(of)g(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))32 b(a)g Fp(c)-5 b(omp)g(act)34 b(subset)41 b Fq(when)34 b(the)e(subspace)i(\()p Fo(A;)17 b Fm(T)3369 1463 y Fj(A)3426 1448 y Fq(\))32 b(is)324 1569 y(a)g(compact)g(space.)44 b(It's)33 b(a)f(n)m(uisance)h(to)f(ha)m (v)m(e)i(to)e(lo)s(ok)f(at)h Fm(T)2585 1584 y Fj(A)2674 1569 y Fq(in)g(order)g(to)g(decide)h(on)324 1689 y(this.)43 b(It)33 b(w)m(ould)f(b)s(e)h(easier)f(to)h(use)g(the)g(original)c Fm(T)d Fq(.)43 b(Thankfully)-8 b(,)33 b(w)m(e)g(can!)324 1858 y Fk(Lemma)k(2.1)49 b Fo(A)41 b Fp(is)h(a)f(c)-5 b(omp)g(act)41 b(subset)g(of)g Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))41 b Fp(i\013)g(every)h Fm(T)25 b Fq(-op)s(en)41 b Fp(c)-5 b(over)41 b(of)h Fo(A)324 1979 y Fp(has)34 b(a)h(\014nite)f(sub)-5 b(c)g(over.)324 2148 y Fq(Pro)s(of)p 324 2161 235 4 v 32 w(Exercise.)324 2317 y Fk(Lemma)37 b(2.2)49 b Fp(Comp)-5 b(actness)37 b(is)h(close)-5 b(d-her)g(e)g (ditary)37 b(and)h(pr)-5 b(eserve)g(d)37 b(by)i(c)-5 b(ontinuous)324 2437 y(maps.)324 2607 y Fq(Pro)s(of)p 324 2620 V 32 w(Exercise.)324 2727 y Fp(Example)324 2847 y Fq(The)33 b(unit)f(circle)g(in)g Fo(R)1174 2811 y Fh(2)1246 2847 y Fq(is)g(compact;)g(indeed,)h(paths)g(in)f(an)m(y)h(space)h(are)f (compact.)324 3130 y Fl(2.1.3)136 b(Compactness)45 b(in)g(Metric)g (Spaces)324 3315 y Fq(In)33 b(an)m(y)g(metric)e(space)j(\()p Fo(M)5 b(;)17 b(d)p Fq(\),)32 b(ev)m(ery)j(compact)d(subset)i Fo(K)40 b Fq(is)32 b(closed)h(and)f(b)s(ounded:)324 3435 y(\(b)s(ounded,)h(since)g(giv)m(en)g(an)m(y)g Fo(x)1520 3450 y Fh(0)1588 3435 y Fm(2)28 b Fo(M)10 b Fq(,)1130 3613 y Fo(K)90 b Fm(\022)83 b Fo(B)5 b Fq(\()p Fo(x)1635 3628 y Fh(0)1675 3613 y Fo(;)17 b Fq(1\))22 b Fm([)g Fo(B)5 b Fq(\()p Fo(x)2088 3628 y Fh(0)2128 3613 y Fo(;)17 b Fq(2\))22 b Fm([)g Fo(B)5 b Fq(\()p Fo(x)2541 3628 y Fh(0)2581 3613 y Fo(;)17 b Fq(3\))22 b Fm([)g(\001)17 b(\001)g(\001)947 3740 y(\))83 b Fo(K)90 b Fm(\022)83 b([)1529 3693 y Fj(j)1529 3763 y(i)p Fh(=1)1648 3740 y Fo(B)5 b Fq(\()p Fo(x)1820 3755 y Fh(0)1860 3740 y Fo(;)17 b(n)1962 3755 y Fj(i)1990 3740 y Fq(\))324 3919 y(where)36 b(w)m(e)g(can)f(arrange)g Fo(n)1348 3934 y Fh(1)1420 3919 y Fo(<)c(n)1585 3934 y Fh(2)1657 3919 y Fo(<)h(:)17 b(:)g(:)31 b(<)h(n)2077 3934 y Fj(j)2113 3919 y Fq(.)51 b(Th)m(us)37 b Fo(K)i Fm(\022)32 b Fo(B)5 b Fq(\()p Fo(x)2844 3934 y Fh(0)2884 3919 y Fo(;)17 b(n)2986 3934 y Fj(j)3023 3919 y Fq(\))35 b(and)g(so)g(an)m(y)324 4039 y(t)m(w)m(o)30 b(p)s(oin)m(ts)e(of)h Fo(K)36 b Fq(lie)28 b(within)g Fo(n)1506 4054 y Fj(j)1572 4039 y Fq(of)g Fo(x)1734 4054 y Fh(0)1803 4039 y Fq(and)i(hence)g(within)e(2)p Fo(n)2664 4054 y Fj(j)2730 4039 y Fq(of)g(eac)m(h)i(other)g(i.e.)41 b Fo(K)324 4159 y Fq(is)32 b(b)s(ounded.)324 4280 y Fo(K)f Fq(is)25 b(closed,)h(since)f(if)e Fo(x)28 b Fm(2)1353 4255 y Fq(\026)1327 4280 y Fo(K)j Fq(and)25 b Fo(x)j Fm(62)g Fo(K)7 b Fq(,)26 b(then)g(for)e(eac)m(h)h Fo(y)31 b Fm(2)d Fo(K)7 b Fq(,)26 b Fo(d)2877 4295 y Fj(y)2946 4280 y Fq(=)3060 4241 y Fh(1)p 3060 4257 36 4 v 3060 4314 a(2)3105 4280 y Fo(d)p Fq(\()p Fo(x;)17 b(y)t Fq(\))26 b Fo(>)i Fq(0)324 4400 y(so)49 b(w)m(e)g(ma)m(y)g(form)e(the)i(\(op)s (en\))g(co)m(v)m(er)h(of)e Fo(K)56 b Fq(as)49 b(follo)m(ws:)74 b Fm(f)p Fo(B)5 b Fq(\()p Fo(y)t(;)17 b(d)2979 4415 y Fj(y)3019 4400 y Fq(\))55 b(:)g Fo(y)j Fm(2)e Fo(K)7 b Fm(g)324 4521 y Fq(whic)m(h)40 b(reduces)h(to)e(a)g(\014nite)h(sub)s (co)m(v)m(er)h Fm(f)p Fo(B)5 b Fq(\()p Fo(y)2058 4536 y Fj(i)2086 4521 y Fo(;)17 b(d)2181 4536 y Fj(y)2216 4546 y Fi(i)2246 4521 y Fq(\))39 b(:)h Fo(y)2438 4536 y Fj(i)2505 4521 y Fm(2)g Fo(K)r(;)17 b(i)39 b Fq(=)g(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)p Fm(g)p Fq(.)64 b(The)324 4641 y(corresp)s(onding)36 b(neigh)m(b)s(ourho)s(o)s(ds)h(of)f Fo(x)p Fq(,)i(namely)e Fo(B)5 b Fq(\()p Fo(x;)17 b(d)2509 4656 y Fj(y)2544 4666 y Fi(i)2574 4641 y Fq(\),)38 b Fo(i)d Fq(=)g(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(n)p Fq(,)38 b(ma)m(y)e(b)s(e)324 4761 y(in)m(tersected)e(giving)d(a)h(neigh)m(b)s (ourho)s(o)s(d)g(of)g Fo(x)h Fq(whic)m(h)g(misses)f Fo(K)40 b Fq(|con)m(tradiction!\))324 4882 y(Neither)32 b(half)g(is)g(v)-5 b(alid)31 b(in)g Fp(al)5 b(l)43 b Fq(top)s(ological)29 b(spaces;)1894 5251 y(15)p eop %%Page: 16 17 16 16 bop 469 548 a Fm(\017)49 b Fq(`compact)23 b Fm(\))g Fq(b)s(ounded')h(do)s(esn't)g(ev)m(en)h(mak)m(e)e(sense)i(since)f(`b)s (ounded')g(dep)s(ends)568 668 y(on)32 b(the)h(metric.)469 989 y Fm(\017)49 b Fq(`compact)34 b Fm(\))g Fq(closed')i(mak)m(es)f (sense)h(but)f(is)g(not)f(alw)m(a)m(ys)i(true.)50 b(F)-8 b(or)34 b(example,)568 1110 y(in)28 b(\()p Fo(R)q(;)17 b Fm(C)6 b Fq(\),)30 b(\(0)p Fo(;)17 b Fq(1\))28 b(is)g(not)h(closed)g (y)m(et)h(it)e Fp(is)37 b Fq(compact)28 b(\(since)i(its)e(top)s(ology)g (is)g(the)568 1230 y(co\014nite)k(top)s(ology!\))324 1426 y(F)-8 b(urther,)29 b(in)f(a)g(metric)f(space,)j(a)e(closed)h(b)s (ounded)g(subset)g(needn't)h(b)s(e)e(compact)g(\(e.g.)324 1546 y(consider)39 b Fo(M)50 b Fq(with)39 b(the)g(discrete)h(metric)e (and)i(let)e Fo(A)h Fm(\022)g Fo(M)50 b Fq(b)s(e)40 b(in\014nite;)h (then)f Fo(A)f Fq(is)324 1666 y(closed,)34 b(b)s(ounded)h(\(since)f Fo(A)c Fm(\022)h Fo(B)5 b Fq(\()p Fo(x;)17 b Fq(2\))30 b(=)g Fo(M)44 b Fq(for)34 b(an)m(y)g Fo(x)d Fm(2)f Fo(M)10 b Fq(\),)35 b(y)m(et)g(it)e(is)g(certainly)324 1787 y(not)28 b(compact!)42 b(Alternativ)m(ely)-8 b(,)28 b(the)h(subspace)h(\(0)p Fo(;)17 b Fq(1\))28 b(is)g(closed)g(\(in)g(itself)7 b(\),)28 b(b)s(ounded,)324 1907 y(but)33 b(not)f(compact.\))324 2028 y(Ho)m(w)m(ev)m(er,)45 b(the)d(Heine-Borel)e(theorem)h(asserts)h (that)f(suc)m(h)i(is)d(the)i(case)g(for)e Fo(R)i Fq(and)324 2148 y Fo(R)399 2112 y Fj(n)446 2148 y Fq(;)32 b(the)h(follo)m(wing)d (is)i(a)h(sp)s(ecial)e(case)j(of)e(the)h(theorem:)324 2343 y Fk(Theorem)k(2.2)49 b Fp(Every)35 b(close)-5 b(d,)34 b(b)-5 b(ounde)g(d)34 b(interval)g Fq([)p Fo(a;)17 b(b)p Fq(])36 b Fp(in)e Fo(R)i Fp(is)f(c)-5 b(omp)g(act.)324 2539 y Fq(Pro)s(of)p 324 2552 235 4 v 41 w(Let)41 b Fm(U)52 b Fq(b)s(e)42 b(an)m(y)g(op)s(en)f(co)m(v)m(er)i(of)e([)p Fo(a;)17 b(b)p Fq(])42 b(and)g(let)f Fo(K)50 b Fq(=)42 b Fm(f)p Fo(x)i Fm(2)f Fq([)p Fo(a;)17 b(b)p Fq(])43 b(:)g([)p Fo(a;)17 b(x)p Fq(])42 b(is)324 2659 y(co)m(v)m(ered)32 b(b)m(y)g(a)e(\014nite)g(subfamily)f(of)h Fm(U)10 b(g)p Fq(.)43 b(Note)31 b(that)f(if)f Fo(x)f Fm(2)g Fo(K)38 b Fq(and)31 b Fo(a)c Fm(\024)i Fo(y)h Fm(\024)f Fo(x)p Fq(,)i(then)324 2780 y Fo(y)40 b Fm(2)d Fo(K)7 b Fq(.)60 b(Clearly)-8 b(,)38 b Fo(K)44 b Fm(6)p Fq(=)37 b Fm(;)h Fq(since)g Fo(a)f Fm(2)g Fo(K)7 b Fq(.)60 b(Moreo)m(v)m(er,)41 b(giv)m(en)d Fo(x)f Fm(2)h Fo(K)7 b Fq(,)39 b(there)g(exists)324 2900 y Fo(\016)367 2915 y Fj(x)448 2900 y Fo(>)d Fq(0)i(suc)m(h)h(that) f([)p Fo(x;)17 b(x)26 b Fq(+)g Fo(\016)1441 2915 y Fj(x)1485 2900 y Fq(\))37 b Fm(\022)g Fo(K)45 b Fq(\(since)38 b Fo(x)g Fm(2)g Fq(some)g(op)s(en)g Fo(U)47 b Fm(2)76 b Fq(c)m(hosen)40 b(\014nite)324 3020 y(sub)s(co)m(v)m(er)34 b(of)e Fm(U)10 b Fq(\).)44 b(Since)33 b Fo(K)39 b Fq(is)33 b(b)s(ounded,)g Fo(k)1972 2984 y Fg(\003)2039 3020 y Fq(=)28 b(sup)17 b Fo(K)40 b Fq(exists.)401 3138 y(\(i\))98 b Fo(k)657 3102 y Fg(\003)724 3138 y Fm(2)29 b Fo(K)p 603 3155 306 4 v 7 w Fq(:)43 b(Cho)s(ose)e Fo(U)51 b Fm(2)41 b(U)51 b Fq(suc)m(h)41 b(that)f Fo(k)2163 3102 y Fg(\003)2243 3138 y Fm(2)h Fo(U)10 b Fq(;)45 b(then)c(there)g(exists) g Fo(\017)f(>)h Fq(0)979 3258 y(suc)m(h)f(that)e(\()p Fo(k)1514 3222 y Fg(\003)1580 3258 y Fm(\000)26 b Fo(\017;)17 b(k)1820 3222 y Fg(\003)1860 3258 y Fq(])37 b Fm(\022)h Fo(U)10 b Fq(.)61 b(Since)39 b(there)g(exists)g Fo(x)f Fm(2)g Fo(K)45 b Fq(suc)m(h)979 3379 y(that)33 b Fo(k)1245 3343 y Fg(\003)1306 3379 y Fm(\000)23 b Fo(\017)28 b(<)f(x)i(<)e(k)1817 3343 y Fg(\003)1857 3379 y Fq(,)32 b Fo(k)1970 3343 y Fg(\003)2037 3379 y Fm(2)c Fo(K)7 b Fq(.)374 3499 y(\(ii\))97 b Fo(k)657 3463 y Fg(\003)724 3499 y Fq(=)28 b Fo(b)p 603 3512 267 4 v 40 w Fq(:)43 b(If)52 b Fo(k)1150 3463 y Fg(\003)1251 3499 y Fo(<)60 b(b)p Fq(,)e(c)m(ho)s(ose)53 b Fo(U)71 b Fm(2)61 b(U)i Fq(with)51 b Fo(k)2526 3463 y Fg(\003)2627 3499 y Fm(2)61 b Fo(U)h Fq(and)53 b(note)f(that)979 3619 y([)p Fo(k)1060 3583 y Fg(\003)1100 3619 y Fo(;)17 b(k)1198 3583 y Fg(\003)1259 3619 y Fq(+)22 b Fo(\016)t Fq(\))28 b Fm(\022)g Fo(U)43 b Fq(for)32 b(some)g Fo(\016)g(>)c Fq(0)k(|con)m(tradiction!)3578 3380 y Fp(Note)324 3739 y Fq(An)d(alternativ)m(e)f(pro)s(of)g([Willard,)f(P)m(age)i(116])f(is)g (to)h(in)m(v)m(ok)m(e)g(the)h(connected)g(nature)f(of)324 3859 y([)p Fo(a;)17 b(b)p Fq(])33 b(b)m(y)g(sho)m(wing)g Fo(K)40 b Fq(is)32 b(clop)s(en)g(in)g([)p Fo(a;)17 b(b)p Fq(].)324 4054 y Fk(Theorem)37 b(2.3)49 b Fp(A)n(ny)25 b(c)-5 b(ontinuous)25 b(map)f(fr)-5 b(om)24 b(a)h(c)-5 b(omp)g(act)24 b(sp)-5 b(ac)g(e)24 b(into)h(a)g(metric)f(sp)-5 b(ac)g(e)324 4175 y(is)34 b(b)-5 b(ounde)g(d.)324 4370 y Fq(Pro)s(of)p 324 4383 235 4 v 32 w(Immediate.)324 4566 y Fk(Corollary)36 b(2.1)49 b Fp(If)38 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))39 b Fp(is)f(c)-5 b(omp)g(act)38 b(and)g Fo(f)46 b Fq(:)34 b Fo(X)43 b Fm(!)34 b Fo(R)40 b Fp(is)e(c)-5 b(ontinuous,)39 b(then)g Fo(f)324 4686 y Fp(is)34 b(b)-5 b(ounde)g(d)35 b(and)f(attains)h(its)g(b)-5 b(ounds.)324 4882 y Fq(Pro)s(of)p 324 4895 V 34 w(Clearly)d(,)36 b Fo(f)46 b Fq(is)35 b(b)s(ounded.)52 b(Let)36 b Fo(m)c Fq(=)h(sup)17 b Fo(f)11 b Fq(\()p Fo(X)d Fq(\))35 b(and)g Fo(l)g Fq(=)d(inf)23 b Fo(f)11 b Fq(\()p Fo(X)d Fq(\);)36 b(w)m(e)g(m)m(ust)324 5002 y(pro)m(v)m(e)49 b(that)f Fo(m)55 b Fm(2)g Fo(f)11 b Fq(\()p Fo(X)d Fq(\))47 b(and)h Fo(l)57 b Fm(2)e Fo(f)11 b Fq(\()p Fo(X)d Fq(\).)89 b(Supp)s(ose)49 b(that)f Fo(m)55 b Fm(62)g Fo(f)11 b Fq(\()p Fo(X)d Fq(\).)89 b(Since)1894 5251 y(16)p eop %%Page: 17 18 17 17 bop 324 463 224 4 v 324 548 a Fo(f)11 b Fq(\()p Fo(X)d Fq(\))31 b(=)g Fo(f)11 b Fq(\()p Fo(X)d Fq(\),)35 b(then)h(there)f(exists)h Fo(\017)c(>)g Fq(0)i(suc)m(h)j(that)d(\()p Fo(m)24 b Fm(\000)g Fo(\017;)17 b(m)25 b Fq(+)e Fo(\017)p Fq(\))h Fm(\\)g Fo(f)11 b Fq(\()p Fo(X)d Fq(\))31 b(=)h Fm(;)324 668 y Fq(i.e.)43 b(for)32 b(all)e Fo(x)e Fm(2)h Fo(X)8 b Fq(,)32 b Fo(f)11 b Fq(\()p Fo(x)p Fq(\))28 b Fm(\024)g Fo(m)22 b Fm(\000)h Fo(\017)17 b(:)g(:)g(:)o Fq(con)m(tra!)324 789 y(Similarly)-8 b(,)29 b(if)j Fo(l)e Fm(62)f Fo(f)11 b Fq(\()p Fo(X)d Fq(\),)33 b(then)g(there)h(exists)g Fo(\017)28 b(>)g Fq(0)33 b(suc)m(h)i(that)d([)p Fo(l)r(;)17 b(l)25 b Fq(+)d Fo(\017)p Fq(\))h Fm(\\)g Fo(f)11 b Fq(\()p Fo(X)d Fq(\))28 b(=)g Fm(;)324 909 y Fq(whence)34 b Fo(l)25 b Fq(+)d Fo(\017)33 b Fq(is)f(a)g(lo)m(w)m(er)h(b)s(ound)f(for)g Fo(f)11 b Fq(\()p Fo(X)d Fq(\)!)324 1198 y Fl(2.1.4)136 b(Sequen)l(tial)46 b(Compactness)324 1383 y Fk(De\014nition)36 b(2.2)49 b Fp(A)33 b(top)-5 b(olo)g(gic)g(al)31 b(sp)-5 b(ac)g(e)31 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))32 b Fp(is)g(said)g(to)g(b)-5 b(e)32 b Fk(sequen)m(tially)h(com-)324 1503 y(pact)i Fp(if)f(and)h(only)f(if)h(every)g(se)-5 b(quenc)g(e)34 b(in)g Fo(X)43 b Fp(has)34 b(a)h(c)-5 b(onver)g(gent)34 b(subse)-5 b(quenc)g(e.)324 1731 y Fq(Recall)29 b(from)h(Chapter)i(1)f(the)g(de\014nition)f(of)h(con)m(v)m (ergence)j(of)c(sequences)35 b(in)30 b(top)s(olog-)324 1852 y(ical)40 b(spaces)j(and)f(the)h(cautionary)e(remarks)h(accompan)m (ying)g(it.)70 b(There)43 b(w)m(e)g(noted)324 1972 y(that,)32 b(con)m(trary)h(to)e(the)i(metric)e(space)i(situation,)e(sequences)k (in)c(top)s(ology)g(can)h(ha)m(v)m(e)324 2093 y Fp(sever)-5 b(al)41 b(di\013er)-5 b(ent)49 b Fq(limits!)64 b(Consider,)43 b(for)c(example,)j(\()p Fo(X)r(;)17 b Fm(T)2624 2108 y Fh(0)2664 2093 y Fq(\))40 b(and)g(\()p Fo(R)q(;)17 b Fm(L)p Fq(\).)67 b(In)40 b(the)324 2213 y(latter)33 b(space,)k(if)c Fo(x)1028 2228 y Fj(n)1106 2213 y Fm(!)e Fo(l)r Fq(,)k(then)h Fo(x)1610 2228 y Fj(n)1688 2213 y Fq(=)31 b Fo(l)37 b Fq(for)d(all)e Fo(n)f Fm(\025)k Fq(some)g Fo(n)2655 2228 y Fh(0)2694 2213 y Fq(.)50 b(Th)m(us)36 b(the)f(sequence)324 2333 y(1)p Fo(;)426 2294 y Fh(1)p 426 2310 36 4 v 426 2368 a(2)471 2333 y Fo(;)525 2294 y Fh(1)p 525 2310 V 525 2368 a(4)570 2333 y Fo(;)624 2294 y Fh(1)p 624 2310 V 624 2368 a(8)669 2333 y Fo(;)17 b(:)g(:)g(:)32 b Fq(do)s(es)h Fp(not)42 b Fq(con)m(v)m(erge)34 b(in)e(\()p Fo(R)q(;)17 b Fm(L)p Fq(\)!)324 2562 y Fk(Lemma)37 b(2.3)49 b Fp(Se)-5 b(quential)47 b(c)-5 b(omp)g(actness)47 b(is)g(close)-5 b(d-her)g(e)g(ditary)47 b(and)h(pr)-5 b(eserve)g(d)46 b(by)324 2682 y(c)-5 b(ontinuous)34 b(maps.)324 2910 y Fq(Pro)s(of)p 324 2923 235 4 v 32 w(Exercise.)324 3031 y(W)-8 b(e)29 b(shall)e(pro)m(v)m(e)i(in)f(the)h(next)g(section)g (that)f(in)g(metric)f(spaces,)k(sequen)m(tial)d(compact-)324 3151 y(ness)34 b(and)e(compactness)i(are)f(equiv)-5 b(alen)m(t!)324 3354 y Fk(De\014nition)36 b(2.3)49 b Fp(Given)36 b(a)h(top)-5 b(olo)g(gic)g(al)35 b(sp)-5 b(ac)g(e)36 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))p Fp(,)37 b(a)g(subset)f Fo(A)h Fp(of)f Fo(X)45 b Fp(and)36 b Fo(x)31 b Fm(2)324 3475 y Fo(X)8 b Fp(,)34 b Fo(x)h Fp(is)f(said)f(to)i(b)-5 b(e)33 b(an)h Fk(accum)m(ulation)h(p)s(oin)m(t)h(of)h(A)d Fp(i\013)g(every)g(neighb) -5 b(ourho)g(o)g(d)33 b(of)324 3595 y Fo(x)i Fp(c)-5 b(ontains)34 b(in\014nitely)h(many)f(p)-5 b(oints)35 b(of)f Fo(A)p Fp(.)324 3824 y Fk(Lemma)j(2.4)49 b Fp(Given)36 b(a)g(c)-5 b(omp)g(act)35 b(sp)-5 b(ac)g(e)36 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))36 b Fp(with)g(an)g(in\014nite)g(subset)g Fo(A)g Fp(of)g Fo(X)8 b Fp(,)324 3944 y(then)34 b Fo(A)h Fq(has)h Fp(an)e(ac)-5 b(cumulation)35 b(p)-5 b(oint.)324 4172 y Fq(Pro)s(of)p 324 4185 V 29 w(Supp)s(ose)31 b(not;)g(then)g(for) f(eac)m(h)h Fo(x)d Fm(2)g Fo(X)8 b Fq(,)30 b(there)h(exists)g(a)f (neigh)m(b)s(ourho)s(o)s(d)f Fo(N)3409 4187 y Fj(x)3483 4172 y Fq(of)324 4293 y Fo(x)36 b Fq(suc)m(h)h(that)e Fo(N)930 4308 y Fj(x)998 4293 y Fm(\\)25 b Fo(A)36 b Fq(is)f(\(at)g(most\))g(\014nite;)i(the)f(family)d Fm(f)p Fo(N)2620 4308 y Fj(x)2697 4293 y Fq(:)g Fo(x)g Fm(2)h Fo(X)8 b Fm(g)35 b Fq(is)g(an)g(op)s(en)324 4413 y(co)m(v)m(er)c(of)e Fo(X)38 b Fq(and)30 b(so)g(has)g(a)g(\014nite)f(sub)s(co)m(v)m(er)j Fm(f)p Fo(N)2127 4428 y Fj(x)2167 4438 y Fi(i)2225 4413 y Fq(:)c Fo(i)g Fq(=)f(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)p Fm(g)p Fq(.)42 b(But)30 b Fo(A)e Fm(\022)g Fo(X)37 b Fq(and)324 4533 y Fo(A)32 b Fq(is)h(in\014nite,)e(whence)1039 4753 y Fo(A)d Fq(=)g Fo(A)22 b Fm(\\)h Fo(X)35 b Fq(=)28 b Fo(A)22 b Fm(\\)g Fq(\()p Fm([)p Fo(N)2013 4768 y Fj(x)2053 4778 y Fi(i)2084 4753 y Fq(\))28 b(=)f Fm([)2319 4712 y Fj(n)2319 4778 y(i)p Fh(=1)2438 4753 y Fq(\()p Fo(A)22 b Fm(\\)h Fo(N)2738 4768 y Fj(x)2778 4778 y Fi(i)2808 4753 y Fq(\))324 4973 y(is)32 b(\014nite!)1894 5251 y(17)p eop %%Page: 18 19 18 18 bop 324 548 a Fk(Lemma)37 b(2.5)49 b Fp(Given)41 b(a)g(se)-5 b(quential)5 b(ly)41 b(c)-5 b(omp)g(act)40 b Fq(metric)h Fp(sp)-5 b(ac)g(e)40 b Fq(\()p Fo(M)5 b(;)17 b(d)p Fq(\))41 b Fp(and)g Fo(\017)f(>)f Fq(0)p Fp(,)324 668 y(ther)-5 b(e)35 b(is)f(a)h Fq(\014nite)g Fp(numb)-5 b(er)34 b(of)h(op)-5 b(en)34 b(b)-5 b(al)5 b(ls,)34 b(r)-5 b(adius)34 b Fo(\017)p Fp(,)i(which)e(c)-5 b(over)34 b Fo(M)10 b Fp(.)324 881 y Fq(Pro)s(of)p 324 894 235 4 v 35 w(Supp)s(ose)37 b(not)e(and)h(that)g(for)f(some)h Fo(\017)d(>)g Fq(0,)k(there)f(exists)h Fp(no)k Fq(\014nite)36 b(family)d(of)324 1002 y(op)s(en)g(balls,)f(radius)g Fo(\017)p Fq(,)i(co)m(v)m(ering)f Fo(M)10 b Fq(.)46 b(W)-8 b(e)33 b(deriv)m(e)h(a)f(con)m(tradiction)f(b)m(y)h(constructing)324 1122 y(a)38 b(sequence)k(\()p Fo(x)915 1137 y Fj(n)962 1122 y Fq(\))d(inductiv)m(ely)f(suc)m(h)i(that)f Fo(d)p Fq(\()p Fo(x)2134 1137 y Fj(m)2200 1122 y Fo(;)17 b(x)2299 1137 y Fj(n)2347 1122 y Fq(\))38 b Fm(\025)g Fo(\017)h Fq(for)g(all)d Fo(n)p Fq(,)41 b Fo(m)e Fq(\()p Fo(n)f Fm(6)p Fq(=)g Fo(m)p Fq(\),)324 1242 y(whence)c(no)f(subsequence)j(is)c (ev)m(en)i(Cauc)m(h)m(y!)324 1363 y(Let)f Fo(x)554 1378 y Fh(1)623 1363 y Fm(2)c Fo(M)44 b Fq(and)34 b(supp)s(ose)g(inductiv)m (ely)f(that)g Fo(x)2183 1378 y Fh(1)2223 1363 y Fo(;)17 b(:)g(:)g(:)f(;)h(x)2497 1378 y Fj(k)2573 1363 y Fq(ha)m(v)m(e)34 b(b)s(een)h(c)m(hosen)f(in)f Fo(M)324 1483 y Fq(suc)m(h)40 b(that)e Fo(d)p Fq(\()p Fo(x)911 1498 y Fj(i)939 1483 y Fo(;)17 b(x)1038 1498 y Fj(j)1074 1483 y Fq(\))38 b Fm(\025)g Fo(\017)g Fq(for)g(all)e Fo(i;)17 b(j)43 b Fm(\024)38 b Fo(k)s(;)17 b(i)38 b Fm(6)p Fq(=)f Fo(j)6 b Fq(.By)39 b(h)m(yp)s(othesis,)i Fm(f)p Fo(B)5 b Fq(\()p Fo(x)3164 1498 y Fj(i)3192 1483 y Fo(;)17 b(\017)p Fq(\))38 b(:)f Fo(i)h Fq(=)324 1604 y(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(k)s Fm(g)28 b Fq(is)g(not)g(an)h(\(op)s(en\))f(co)m(v)m(er)i(of)e Fo(M)39 b Fq(and)28 b(so)h(there)g(exists)g Fo(x)2779 1619 y Fj(k)r Fh(+1)2940 1604 y Fm(2)f Fo(M)39 b Fq(suc)m(h)30 b(that)324 1724 y Fo(d)p Fq(\()p Fo(x)468 1739 y Fj(k)r Fh(+1)601 1724 y Fo(;)17 b(x)700 1739 y Fj(i)728 1724 y Fq(\))28 b Fm(\025)g Fo(\017)33 b Fq(for)e(1)d Fm(\024)g Fo(i)g Fm(\024)g Fo(k)s Fq(.)44 b(W)-8 b(e)32 b(th)m(us)i(construct)f (the)g(required)g(sequence)i(\()p Fo(x)3449 1739 y Fj(n)3497 1724 y Fq(\),)324 1844 y(whic)m(h)e(clearly)f(has)g(no)h(con)m(v)m (ergen)m(t)i(subsequence.)324 2036 y Fk(Theorem)i(2.4)49 b Fp(A)35 b(metric)g(sp)-5 b(ac)g(e)34 b(is)g(c)-5 b(omp)g(act)34 b(i\013)h(it)g(is)g(se)-5 b(quential)5 b(ly)34 b(c)-5 b(omp)g(act.)324 2228 y Fq(Pro)s(of)p 324 2241 V 392 2420 a Fm(\))p Fq(:)49 b(Supp)s(ose)30 b(\()p Fo(M)5 b(;)17 b(d)p Fq(\))29 b(is)h(compact.)42 b(Giv)m(en)29 b(an)m(y)i(sequence)h(\()p Fo(x)2724 2435 y Fj(n)2771 2420 y Fq(\))e(in)f Fo(M)10 b Fq(,)31 b(either)e Fo(A)f Fq(=)568 2540 y Fm(f)p Fo(x)673 2555 y Fh(1)712 2540 y Fo(;)17 b(x)811 2555 y Fh(2)851 2540 y Fo(;)g(:)g(:)g(:)o Fm(g)27 b Fq(is)g(\014nite)g(or)f(it)g(is)h(in\014nite.)41 b(If)27 b Fo(A)g Fq(is)g(\014nite,)h(there)g(m)m(ust)f(b)s(e)g(at)g (least)568 2661 y(one)37 b(p)s(oin)m(t)g Fo(l)i Fq(in)e Fo(A)g Fq(whic)m(h)h(o)s(ccurs)g(in\014nitely)e(often)h(in)g(the)h (sequence)i(and)d(its)568 2781 y(o)s(ccurrences)27 b(form)d(a)h (subsequence)j(con)m(v)m(erging)e(to)f Fo(l)r Fq(.)41 b(If)25 b Fo(A)g Fq(is)g(in\014nite,)h(then)g(b)m(y)568 2902 y(the)31 b(previous)g(lemma)e(there)i(exists)h Fo(x)c Fm(2)g Fo(X)38 b Fq(suc)m(h)32 b(that)f(ev)m(ery)i(neigh)m(b)s(ourho)s (o)s(d)568 3022 y(of)f Fo(x)h Fq(con)m(tains)f(in\014nitely)f(man)m(y)i (p)s(oin)m(ts)f(of)g Fo(A)p Fq(.)568 3182 y(F)-8 b(or)34 b(eac)m(h)i Fo(k)g Fm(2)d Fo(!)t Fq(,)i Fo(B)5 b Fq(\()p Fo(x;)1508 3143 y Fh(1)p 1506 3159 39 4 v 1506 3216 a Fj(k)1555 3182 y Fq(\))35 b(con)m(tains)g(in\014nitely)f(man)m(y)h Fo(x)2748 3197 y Fj(n)2796 3182 y Fq('s:)49 b(select)36 b(one,)g(call)568 3302 y(it)31 b Fo(x)720 3317 y Fj(n)763 3329 y Fi(k)805 3302 y Fq(,)h(making)f(sure)h(that)g Fo(n)1682 3317 y Fj(k)1753 3302 y Fo(>)27 b(n)1914 3317 y Fj(k)r Fg(\000)p Fh(1)2075 3302 y Fo(>)g(n)2236 3317 y Fj(k)r Fg(\000)p Fh(2)2386 3302 y Fo(:)17 b(:)g(:)p Fq(.)43 b(W)-8 b(e)32 b(ha)m(v)m(e)h(a)f(subsequence)568 3423 y(\()p Fo(x)661 3438 y Fj(n)704 3447 y Fe(1)743 3423 y Fo(;)17 b(x)842 3438 y Fj(n)885 3447 y Fe(2)923 3423 y Fo(;)g(:)g(:)g(:)f(;)h(x)1197 3438 y Fj(n)1240 3450 y Fi(k)1282 3423 y Fo(;)g(:)g(:)g(:)p Fq(\))38 b(so)h(that)f Fo(d)p Fq(\()p Fo(x;)17 b(x)2103 3438 y Fj(n)2146 3450 y Fi(k)2189 3423 y Fq(\))38 b Fo(<)2390 3383 y Fh(1)p 2389 3399 V 2389 3457 a Fj(k)2475 3423 y Fm(!)g Fq(0)g(i.e.)62 b Fo(x)2942 3438 y Fj(n)2985 3450 y Fi(k)3065 3423 y Fm(!)38 b Fo(x)p Fq(.)62 b(Th)m(us)568 3543 y(in)39 b(either)h(case)h (there)f(exists)h(a)f(con)m(v)m(ergen)m(t)i(subsequence)h(and)d(so)h (\()p Fo(M)5 b(;)17 b(d)p Fq(\))39 b(is)568 3663 y(sequen)m(tially)32 b(compact.)392 3863 y Fm(\()p Fq(:)49 b(Con)m(v)m(ersely)-8 b(,)40 b(supp)s(ose)e(\()p Fo(M)5 b(;)17 b(d)p Fq(\))37 b(is)f(sequen)m(tially)h(compact)f(and)h Fp(not)47 b Fq(compact.)568 3983 y(Then)36 b(there)f(exists)h(some)e(op)s(en)h(co)m (v)m(er)i Fm(f)p Fo(G)2214 3998 y Fj(i)2273 3983 y Fq(:)32 b Fo(i)g Fm(2)f Fo(I)8 b Fm(g)35 b Fq(of)f Fo(M)46 b Fq(ha)m(ving)34 b Fp(no)41 b Fq(\014nite)568 4104 y(sub)s(co)m(v)m(er.) 60 b(By)38 b(Lemma)e(2.5,)j(with)e Fo(\017)f Fq(=)2152 4065 y Fh(1)p 2148 4081 43 4 v 2148 4138 a Fj(n)2238 4104 y Fq(\()p Fo(n)g Fm(2)h Fo(!)t Fq(\),)h(w)m(e)h(can)e(co)m(v)m(er) i Fo(M)48 b Fq(b)m(y)39 b(a)568 4224 y Fp(\014nite)k Fq(n)m(um)m(b)s(er)36 b(of)g(balls)f(of)g(radius)1958 4185 y Fh(1)p 1954 4201 V 1954 4259 a Fj(n)2007 4224 y Fq(.)54 b(F)-8 b(or)35 b(eac)m(h)i Fo(n)p Fq(,)g(there)g(has)g(to)e (b)s(e)i(one)f(of)568 4345 y(these,)i(sa)m(y)f Fo(B)5 b Fq(\()p Fo(x)1193 4360 y Fj(n)1241 4345 y Fo(;)1298 4305 y Fh(1)p 1294 4321 V 1294 4379 a Fj(n)1347 4345 y Fq(\),)37 b(whic)m(h)g(cannot)f(b)s(e)h(co)m(v)m(ered)h(b)m(y)f(an)m (y)g(\014nite)f(n)m(um)m(b)s(er)g(of)568 4465 y(the)k(sets)h Fo(G)1018 4480 y Fj(i)1047 4465 y Fq(.)65 b(The)41 b(sequence)i(\()p Fo(x)1852 4480 y Fj(n)1900 4465 y Fq(\))c(m)m(ust)i(ha)m(v)m(e)g(a)f (con)m(v)m(ergen)m(t)i(subsequence)568 4585 y(\()p Fo(x)661 4600 y Fj(n)704 4612 y Fi(k)746 4585 y Fq(\))j(whic)m(h)g(con)m(v)m (erges)i(to)e(a)g(limit)c Fo(l)r Fq(.)81 b(Y)-8 b(et)45 b Fm(f)p Fo(G)2497 4600 y Fj(i)2573 4585 y Fq(:)k Fo(i)g Fm(2)g Fo(I)8 b Fm(g)45 b Fq(co)m(v)m(ers)i Fo(M)10 b Fq(,)48 b(so)568 4706 y Fo(l)30 b Fm(2)60 b Fq(some)33 b Fo(G)1075 4721 y Fj(i)1099 4730 y Fe(0)1137 4706 y Fq(,)g(sa)m(y)-8 b(.)568 4866 y(As)30 b Fo(k)h Fm(!)c(1)p Fq(,)j Fo(x)1130 4881 y Fj(n)1173 4893 y Fi(k)1244 4866 y Fm(!)d Fo(l)r Fq(;)k(but)f(also)1862 4826 y Fh(1)p 1839 4842 81 4 v 1839 4900 a Fj(n)1882 4912 y Fi(k)1958 4866 y Fm(!)d Fq(0)j(and)g(1)p Fo(=n)2507 4881 y Fj(k)2579 4866 y Fq(is)g(the)g(radius)g(of)f(the)h(ball)568 5002 y(cen)m(tred)g(on)f Fo(x)1095 5017 y Fj(n)1138 5029 y Fi(k)1180 5002 y Fq(.)43 b(So)29 b(ev)m(en)m(tually)g Fo(B)5 b Fq(\()p Fo(x)2016 5017 y Fj(n)2059 5029 y Fi(k)2102 5002 y Fo(;)2178 4963 y Fh(1)p 2155 4979 V 2155 5036 a Fj(n)2198 5048 y Fi(k)2246 5002 y Fq(\))29 b(is)f(inside)h Fo(G)2758 5017 y Fj(i)2782 5026 y Fe(0)2820 5002 y Fq(,)h(con)m (tradictory)f(to)1894 5251 y(18)p eop %%Page: 19 20 19 19 bop 568 548 a Fq(their)32 b(c)m(hoice!)44 b(\(More)32 b(rigorously)-8 b(,)31 b(there)j(exists)f Fo(m)28 b Fm(2)g Fo(!)36 b Fq(suc)m(h)e(that)e Fo(B)5 b Fq(\()p Fo(l)r(;)3360 509 y Fh(2)p 3346 525 63 4 v 3346 582 a Fj(m)3419 548 y Fq(\))27 b Fm(\022)568 668 y Fo(G)645 683 y Fj(i)669 692 y Fe(0)707 668 y Fq(.)70 b(No)m(w)42 b Fo(B)5 b Fq(\()p Fo(l)r(;)1250 629 y Fh(1)p 1237 645 V 1237 703 a Fj(m)1309 668 y Fq(\))41 b(con)m(tains)g Fo(x)1834 683 y Fj(n)1877 695 y Fi(k)1961 668 y Fq(for)g(all)e Fo(k)45 b Fm(\025)e Fo(k)2530 683 y Fh(0)2610 668 y Fq(sa)m(y)-8 b(,)45 b(so)c(c)m(ho)s (ose)h Fo(k)k Fm(\025)c Fo(k)3522 683 y Fh(0)568 789 y Fq(suc)m(h)e(that)e Fo(n)1069 804 y Fj(k)1150 789 y Fm(\025)h Fo(m)p Fq(.)62 b(Then)40 b Fo(B)5 b Fq(\()p Fo(x)1873 804 y Fj(n)1916 816 y Fi(k)1959 789 y Fo(;)2035 749 y Fh(1)p 2013 765 81 4 v 2013 823 a Fj(n)2056 835 y Fi(k)2103 789 y Fq(\))39 b Fm(\022)f Fo(B)5 b Fq(\()p Fo(l)r(;)2511 749 y Fh(2)p 2497 765 63 4 v 2497 823 a Fj(m)2570 789 y Fq(\))38 b Fm(\022)g Fo(G)2838 804 y Fj(i)2862 813 y Fe(0)2901 789 y Fq(.\))62 b(Hence,)42 b Fo(M)49 b Fq(is)568 909 y(compact.)324 1197 y Fl(2.1.5)136 b(Compactness)45 b(and)g(Uniform)h(Con)l(tin)l(uit)l(y)324 1382 y Fq(Recall)36 b(that)h(a)g(map)g Fo(f)47 b Fq(:)37 b(\()p Fo(X)1425 1397 y Fh(1)1464 1382 y Fo(;)17 b(d)1559 1397 y Fh(1)1598 1382 y Fq(\))36 b Fm(!)g Fq(\()p Fo(X)1927 1397 y Fh(2)1966 1382 y Fo(;)17 b(d)2061 1397 y Fh(2)2100 1382 y Fq(\),)39 b(where)f(\()p Fo(X)2609 1397 y Fj(i)2638 1382 y Fo(;)17 b(d)2733 1397 y Fj(i)2760 1382 y Fq(\))38 b(is)f(a)g(metric)f(space)324 1502 y(for)28 b(eac)m(h)i Fo(i)p Fq(,)h(is)d Fp(uniformly)k(c)-5 b(ontinuous)31 b(on)g Fo(X)2016 1517 y Fj(i)2074 1502 y Fq(if)d(giv)m(en)h(an)m(y)h Fo(\017)e(>)f Fq(0)p Fo(;)17 b Fm(9)p Fo(\016)32 b(>)27 b Fq(0)i(suc)m(h)i(that)324 1623 y Fo(d)375 1638 y Fh(1)414 1623 y Fq(\()p Fo(x;)17 b(y)t Fq(\))27 b Fo(<)g(\016)37 b Fq(for)32 b Fo(x;)17 b(y)31 b Fm(2)d Fo(X)1353 1638 y Fh(1)1420 1623 y Fm(\))f Fo(d)1598 1638 y Fh(2)1637 1623 y Fq(\()p Fo(f)11 b Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(f)11 b Fq(\()p Fo(y)t Fq(\)\))26 b Fo(<)i(\017:)324 1743 y Fq(Ordinary)40 b(con)m(tin)m(uit)m(y)h(of)f Fo(f)51 b Fq(is)41 b(a)f(lo)s(cal)f(prop)s(ert)m(y)-8 b(,)43 b(while)d(uniform)f(con)m(tin)m(uit)m(y)i(is)f(a)324 1863 y(global)e(prop)s(ert)m(y)j(since)g(it)f(sa)m(ys)i(something)e(ab) s(out)g(the)h(b)s(eha)m(viour)f(of)g Fo(f)52 b Fq(o)m(v)m(er)41 b(the)324 1984 y(whole)36 b(space)i Fo(X)950 1999 y Fh(1)989 1984 y Fq(.)56 b(Since)36 b(compactness)i(allo)m(ws)e(us)h(to)f(pass)h (from)f(the)h(lo)s(cal)d(to)i(the)324 2104 y(global,)30 b(the)j(next)h(result)e(is)g(not)h(surprising:)324 2303 y Fk(Theorem)k(2.5)49 b Fp(If)31 b Fq(\()p Fo(X)r(;)17 b(d)p Fq(\))31 b Fp(is)h(a)f(c)-5 b(omp)g(act)31 b(metric)g(sp)-5 b(ac)g(e)31 b(and)g Fo(f)39 b Fq(:)27 b Fo(X)36 b Fm(!)27 b Fo(R)33 b Fp(is)e(c)-5 b(ontin-)324 2423 y(uous,)35 b(then)g Fo(f)45 b Fp(is)35 b(uniformly)f(c)-5 b(ontinuous)35 b(on)f Fo(X)8 b Fp(.)324 2622 y(Note)33 b Fq(Result)f(holds)g(for)g(an) m(y)i(metric)d(space)j(co)s(domain.)324 2742 y(Pro)s(of)p 324 2755 235 4 v 37 w(Let)k Fo(\017)g(>)e Fq(0;)41 b(since)d Fo(f)49 b Fq(is)37 b(con)m(tin)m(uous,)j(for)e(eac)m(h)h Fo(x)e Fm(2)g Fo(X)8 b Fq(,)40 b Fm(9)p Fo(\016)2877 2757 y Fj(x)2958 2742 y Fo(>)d Fq(0)g(suc)m(h)j(that)324 2863 y Fo(d)p Fq(\()p Fo(x;)17 b(y)t Fq(\))28 b Fo(<)i Fq(2)p Fo(\016)828 2878 y Fj(x)905 2863 y Fm(\))f(j)p Fo(f)11 b Fq(\()p Fo(x)p Fq(\))23 b Fm(\000)g Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fm(j)28 b Fo(<)1737 2823 y Fj(\017)p 1734 2839 36 4 v 1734 2897 a Fh(2)1779 2863 y Fq(.)47 b(The)34 b(family)e Fm(f)p Fo(B)2478 2878 y Fj(\016)2509 2886 y Fi(x)2552 2863 y Fq(\()p Fo(x)p Fq(\))e(:)f Fo(x)h Fm(2)g Fo(X)8 b Fm(g)34 b Fq(is)f(an)g(op)s(en)324 2983 y(co)m(v)m(er)38 b(of)e Fo(X)45 b Fq(and)36 b(so)h(has)h(a)e(\014nite)g (sub)s(co)m(v)m(er)j Fm(f)p Fo(B)2185 2998 y Fj(\016)2216 3006 y Fi(x)2251 3022 y(i)2286 2983 y Fq(\()p Fo(x)2379 2998 y Fj(i)2407 2983 y Fq(\))c(:)g Fo(i)g Fq(=)f(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)p Fm(g)36 b Fq(of)h Fo(X)8 b Fq(.)55 b(Let)324 3103 y Fo(\016)41 b Fq(=)c(min)o Fm(f)p Fo(\016)777 3118 y Fj(x)817 3128 y Fi(i)885 3103 y Fq(:)g Fo(i)h Fq(=)f(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)p Fm(g)p Fq(;)41 b(then,)f(giv)m(en)f Fo(x;)17 b(y)41 b Fm(2)c Fo(X)46 b Fq(suc)m(h)40 b(that)e Fo(d)p Fq(\()p Fo(x;)17 b(y)t Fq(\))37 b Fo(<)g(\016)t Fq(,)j(it)324 3224 y(follo)m(ws)31 b(that)h Fm(j)p Fo(f)11 b Fq(\()p Fo(x)p Fq(\))22 b Fm(\000)h Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fm(j)26 b Fo(<)i(\017)324 3344 y Fq(\(for)h Fo(x)f Fm(2)g Fo(B)759 3359 y Fj(\016)790 3367 y Fi(x)825 3383 y(i)860 3344 y Fq(\()p Fo(x)953 3359 y Fj(i)982 3344 y Fq(\))h(for)h(some)f Fo(i)p Fq(,)i(whence)h Fo(d)p Fq(\()p Fo(x;)17 b(x)2110 3359 y Fj(i)2138 3344 y Fq(\))28 b Fo(<)f(\016)2350 3359 y Fj(x)2390 3369 y Fi(i)2450 3344 y Fq(and)j(so)g Fo(d)p Fq(\()p Fo(y)t(;)17 b(x)2994 3359 y Fj(i)3022 3344 y Fq(\))27 b Fm(\024)h Fo(d)p Fq(\()p Fo(y)t(;)17 b(x)p Fq(\))g(+)324 3465 y Fo(d)p Fq(\()p Fo(x;)g(x)567 3480 y Fj(i)595 3465 y Fq(\))28 b Fo(<)f(\016)f Fq(+)c Fo(\016)974 3480 y Fj(x)1014 3490 y Fi(i)1073 3465 y Fm(\024)28 b Fq(2)p Fo(\016)1270 3480 y Fj(x)1310 3490 y Fi(i)1368 3465 y Fm(\))f(j)p Fo(f)11 b Fq(\()p Fo(y)t Fq(\))20 b Fm(\000)j Fo(f)11 b Fq(\()p Fo(x)1982 3480 y Fj(i)2010 3465 y Fq(\))p Fm(j)28 b Fo(<)2220 3425 y Fj(\017)p 2217 3441 V 2217 3499 a Fh(2)2262 3465 y Fq(.)324 3585 y(Th)m(us)34 b Fm(j)p Fo(f)11 b Fq(\()p Fo(x)p Fq(\))22 b Fm(\000)g Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fm(j)26 b(\024)j(j)p Fo(f)11 b Fq(\()p Fo(x)p Fq(\))21 b Fm(\000)i Fo(f)11 b Fq(\()p Fo(x)1748 3600 y Fj(i)1776 3585 y Fq(\))p Fm(j)22 b Fq(+)g Fm(j)p Fo(f)11 b Fq(\()p Fo(x)2142 3600 y Fj(i)2170 3585 y Fq(\))22 b Fm(\000)h Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fm(j)26 b Fo(<)2688 3546 y Fj(\017)p 2684 3562 V 2684 3619 a Fh(2)2752 3585 y Fq(+)2863 3546 y Fj(\017)p 2860 3562 V 2860 3619 a Fh(2)2933 3585 y Fq(=)h Fo(\017)p Fq(\).)324 3705 y Fp(Note)39 b Fq(Compactness)h(is)e(not)h(a)g(necessary)i (condition)c(on)i(the)g(domain)e(for)h(uniform)324 3826 y(con)m(tin)m(uit)m(y)-8 b(.)47 b(F)-8 b(or)34 b(example,)g(for)f Fp(any)43 b Fq(metric)32 b(space)k(\()p Fo(X)r(;)17 b(d)p Fq(\),)34 b(let)f Fo(f)41 b Fq(:)30 b Fo(X)38 b Fm(!)29 b Fo(X)42 b Fq(b)s(e)34 b(the)324 3946 y(iden)m(tit)m(y)e(map.)43 b(Then)34 b Fo(f)43 b Fq(is)32 b(easily)g(seen)i(to)e(b)s(e)h (uniformly)d(con)m(tin)m(uous)j(on)g Fo(X)8 b Fq(.)324 4234 y Fl(2.1.6)136 b(Lo)t(cal)45 b(Compactness)324 4419 y Fk(De\014nition)36 b(2.4)49 b Fp(A)26 b(top)-5 b(olo)g(gic)g(al)25 b(sp)-5 b(ac)g(e)25 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))h Fp(is)f Fk(lo)s(cally)f(compact)h Fp(i\013)h(e)-5 b(ach)25 b(p)-5 b(oint)324 4539 y(of)34 b Fo(X)43 b Fp(has)34 b(a)h(c)-5 b(omp)g(act)34 b(neighb)-5 b(ourho)g(o)g(d.)324 4761 y Fq(Clearly)d(,)41 b(ev)m(ery)h(compact)e(space)h(is)f(lo)s (cally)d(compact.)66 b(Ho)m(w)m(ev)m(er,)45 b(the)40 b(con)m(v)m(erse)j(is)324 4882 y Fp(not)f Fq(true.)324 5002 y Fp(Examples)1894 5251 y Fq(19)p eop %%Page: 20 21 20 20 bop 416 548 a Fq(\(i\))48 b(With)22 b Fo(X)31 b Fq(in\014nite,)24 b(the)g(discrete)f(space)i(\()p Fo(X)r(;)17 b Fm(D)s Fq(\))22 b(is)h(clearly)f(lo)s(cally)e(compact)j(\(for)568 668 y(eac)m(h)33 b Fo(x)28 b Fm(2)g Fo(X)8 b Fq(,)33 b Fm(f)p Fo(x)p Fm(g)f Fq(is)g(a)h(compact)f(neigh)m(b)s(ourho)s(o)s(d) g(of)g Fo(x)p Fq(!\))43 b(but)33 b Fp(not)42 b Fq(compact.)389 858 y(\(ii\))47 b(With)f Fo(X)54 b Fq(in\014nite)46 b(and)g Fo(x)52 b Fm(2)g Fo(X)8 b Fq(,)50 b(\()p Fo(X)r(;)17 b Fm(I)7 b Fq(\()p Fo(x)p Fq(\)\))48 b(is)e(lo)s(cally)e(compact)i (\(but)h(not)568 978 y(compact\).)362 1167 y(\(iii\))f(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\))35 b(is)g(lo)s(cally)e(compact)j(\()p Fo(x)d Fm(2)g Fo(R)h Fm(\))e Fq([)p Fo(x)25 b Fm(\000)f Fq(1)p Fo(;)17 b(x)25 b Fq(+)f(1])35 b(is)g(a)g(compact)h(neigh-)568 1288 y(b)s(ourho)s(o)s(d)31 b(of)h Fo(x)p Fq(\).)364 1477 y(\(iv\))49 b(The)34 b(set)h(of)e(rational)e(n)m(um)m(b)s(ers)k Fo(Q)f Fq(with)f(its)g(usual)g(top)s(ology)g(is)g Fp(not)43 b Fq(a)33 b(lo)s(cally)568 1597 y(compact)25 b(space,)j(for)d(supp)s (ose)i(otherwise;)i(then)d(0)g(has)g(a)f(compact)g(neigh)m(b)s(our-)568 1718 y(ho)s(o)s(d)36 b Fo(C)45 b Fq(in)36 b Fo(Q)i Fq(so)f(w)m(e)i(can) f(c)m(ho)s(ose)g Fo(\017)e(>)g Fq(0)h(suc)m(h)i(that)e 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Fq(\))39 b Fp(which)e(is)h(c)-5 b(ontinuous)38 b(and)g(onto;)h Fq(\()p Fo(Q;)17 b Fm(D)s Fq(\))38 b Fp(is)g(lo)-5 b(c)g(al)5 b(ly)38 b(c)-5 b(omp)g(act)37 b(while)568 2680 y Fq(\()p Fo(Q;)17 b Fm(I)781 2695 y Fj(Q)840 2680 y Fq(\))35 b Fp(isn)-10 b('t!)324 2852 y Fq(Pro)s(of)p 324 2865 235 4 v 32 w(Exercise.)324 3178 y Fn(2.2)160 b(Other)53 b(Co)l(v)l(ering)f (Conditions)324 3397 y Fk(De\014nition)36 b(2.5)49 b Fp(A)35 b(top)-5 b(olo)g(gic)g(al)34 b(sp)-5 b(ac)g(e)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))34 b Fp(is)h(said)f(to)h(b)-5 b(e)409 3558 y(\(i\))49 b Fk(Lindel\177)-56 b(of)34 b Fp(i\013)h(every)f(op)-5 b(en)35 b(c)-5 b(over)34 b(of)g Fo(X)43 b Fp(has)34 b(a)h(c)-5 b(ountable)35 b(sub)-5 b(c)g(over)379 3747 y(\(ii\))49 b Fk(coun)m(tably)40 b(compact)c Fp(i\013)h(every)g(c)-5 b(ountable)37 b(op)-5 b(en)36 b(c)-5 b(over)37 b(of)g Fo(X)44 b Fp(has)37 b(a)g(\014nite)568 3867 y(sub)-5 b(c)g(over.)324 4039 y Fq(Th)m(us,)41 b(a)d(space)h(is)f (compact)g(precisely)g(when)i(it)d(is)g(b)s(oth)h(Lindel\177)-49 b(of)37 b(and)h(coun)m(tably)324 4159 y(compact.)59 b(F)-8 b(urther,)39 b(ev)m(ery)h(sequen)m(tially)d(compact)h(space)g(is)g (coun)m(tably)g(compact,)324 4280 y(although)30 b(the)h(con)m(v)m(erse) i(is)e(not)f(true.)44 b(Moreo)m(v)m(er,)32 b(sequen)m(tial)g (compactness)g(neither)324 4400 y(implies)e(nor)i(is)g(implied)e(b)m(y) k(compactness.)324 4521 y(Ho)m(w)m(ev)m(er,)i(for)d(metric)g(spaces,)i (or)e(more)g(generally)-8 b(,)33 b(metrizable)f(spaces,)j(the)g(condi-) 324 4641 y(tions)d(compact,)g(coun)m(tably)h(compact)f(and)h(sequen)m (tially)f(compact)g(are)h(equiv)-5 b(alen)m(t.)324 4761 y Fp(Note)24 b Fq(Second)h(coun)m(table)f Fm(\))g Fq(separable;)j (separable)d(+)f(metrizable)g Fm(\))g Fq(second)i(coun)m(t-)324 4882 y(able)k(.)16 b(.)g(.)g(and)31 b(so)g(in)e(metrizable)f(spaces,)33 b(second)e(coun)m(tabilit)m(y)e(and)h(separabilit)m(y)f(are)324 5002 y(equiv)-5 b(alen)m(t.)1894 5251 y(20)p eop %%Page: 21 22 21 21 bop 324 548 a Fn(2.3)160 b(Connectedness)324 767 y Fq(It)37 b(is)g(not)g(terribly)f(hard)i(to)f(kno)m(w)h(when)h(a)e (set)h(on)f(the)h(real)e(line)g(is)h(connected,)j(or)324 887 y(`of)k(just)i(one)f(piece.')82 b(This)45 b(notion)f(is)g(extended) j(to)e(general)g(top)s(ological)c(spaces)324 1008 y(in)36 b(this)g(section)h(and)f(alternativ)m(e)g(c)m(haracterizations)g(of)g (the)h(notion)f(are)h(giv)m(en.)55 b(In)324 1128 y(addition)38 b(the)i(relationship)d(b)s(et)m(w)m(een)42 b(con)m(tin)m(uous)f(maps)e (and)h(and)f(connectedness)324 1249 y(is)d(giv)m(en.)58 b(This)37 b(pro)m(vides)h(an)f(elegan)m(t)g(restatemen)m(t)h(of)e(the)i (familiar)33 b(In)m(termediate)324 1369 y(V)-8 b(alue)32 b(Theorem)h(from)e(\014rst)i(term)f(calculus.)324 1657 y Fl(2.3.1)136 b(De\014nition)45 b(of)h(Connectedness)324 1842 y Fq(A)39 b Fp(p)-5 b(artition)46 b Fq(of)39 b(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))39 b(means)g(a)g(pair)f(of)g(disjoin)m (t,)i(non-empt)m(y)-8 b(,)41 b Fm(T)26 b Fq(-op)s(en)39 b(subsets)324 1962 y(whose)47 b(union)f(is)g Fo(X)8 b Fq(.)84 b(Notice)46 b(that,)j(since)e(these)g(sets)h(are)e(complemen)m (ts)g(of)f(one)324 2083 y(another,)32 b(they)g(are)g(b)s(oth)f(closed)h (as)f(w)m(ell)g(as)h(b)s(oth)f(op)s(en.)43 b(Indeed,)34 b(the)e(de\014nition)e(of)324 2203 y('partition')g(is)j(not)f (a\013ected)h(b)m(y)h(replacing)d(the)i(term)f('op)s(en')h(b)m(y)h ('closed'.)324 2404 y Fk(De\014nition)i(2.6)49 b Fp(A)41 b Fk(connected)g Fp(sp)-5 b(ac)g(e)39 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))40 b Fp(is)h(one)f(which)f(has)h Fq(no)h Fp(p)-5 b(artition.)324 2524 y(\(Otherwise,)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))35 b Fp(is)g(said)f(to)h(b)-5 b(e)34 b Fk(disconnected)p Fp(.\))324 2645 y(If)40 b Fm(;)g(6)p Fq(=)f Fo(A)g Fm(\022)h Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))p Fp(,)43 b(we)d(c)-5 b(al)5 b(l)41 b Fo(A)g Fp(a)g(c)-5 b(onne)g(cte)g(d)41 b(set)g(in)g Fo(X)49 b Fp(whenever)40 b Fq(\()p Fo(A;)17 b Fm(T)3265 2660 y Fj(A)3322 2645 y Fq(\))41 b Fp(is)g(a)324 2765 y(c)-5 b(onne)g(cte)g(d)34 b(sp)-5 b(ac)g(e.)324 2989 y Fk(Lemma)37 b(2.7)49 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))35 b Fp(is)e(c)-5 b(onne)g(cte)g(d)33 b(i\013)h Fo(X)42 b Fp(and)34 b Fm(;)g Fp(ar)-5 b(e)34 b(the)g(only)g(subsets)g (which)f(ar)-5 b(e)324 3110 y(clop)g(en.)324 3369 y Fk(Examples)416 3554 y Fq(\(i\))48 b(\()p Fo(X)r(;)17 b Fm(T)787 3569 y Fh(0)826 3554 y Fq(\))33 b(is)f(connected.)389 3877 y(\(ii\))47 b(\()p Fo(X)r(;)17 b Fm(D)s Fq(\))23 b(cannot)h(b)s(e)h (connected)g(unless)g Fm(j)p Fo(X)8 b Fm(j)27 b Fq(=)g(1.)41 b(\(Indeed)25 b(the)f(only)g(connected)568 3997 y(subsets)34 b(are)f(the)g(singletons!\))362 4320 y(\(iii\))46 b(The)33 b(Sorgenfrey)h(line)d Fo(R)1509 4335 y Fj(s)1578 4320 y Fq(is)h(disconnected)i(\(for)e([)p Fo(x;)17 b Fm(1)p Fq(\))33 b(is)f(clop)s(en!\).)364 4643 y(\(iv\))49 b(The)33 b(subspace)i Fo(Q)d Fq(of)h(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\))32 b(is)g(not)h(connected)h(b)s(ecause)1361 4860 y Fo(Q)23 b Fm(\\)f Fq([)p Fm(\000)1653 4773 y(p)p 1737 4773 49 4 v 1737 4860 a Fq(2)p Fo(;)1830 4773 y Fm(p)p 1912 4773 V 1912 4860 a Fq(2])1361 4919 y Ff(|)p 1398 4919 239 10 v 239 w({z)p 1711 4919 V 239 w(})1446 5027 y Fq(closed)33 b(in)f Fj(Q)2016 4860 y Fq(=)27 b Fo(Q)c Fm(\\)f Fq(\()p Fm(\000)2422 4773 y(p)p 2506 4773 49 4 v 2506 4860 a Fq(2)p Fo(;)2599 4773 y Fm(p)p 2681 4773 V 2681 4860 a Fq(2\))2119 4919 y Ff(|)p 2156 4919 250 10 v 250 w({z)p 2480 4919 V 250 w(})2241 5024 y Fq(op)s(en)33 b(in)f Fj(Q)1894 5251 y Fq(21)p eop %%Page: 22 23 22 22 bop 568 548 a Fq(is)32 b(clop)s(en)g(and)h(is)f(neither)g(univ)m (ersal)h(nor)f(empt)m(y)-8 b(.)392 870 y(\(v\))49 b(\()p Fo(X)r(;)17 b Fm(C)6 b Fq(\))31 b(is)g(connected)i(except)g(when)f Fo(X)39 b Fq(is)31 b(\014nite;)h(indeed,)g(ev)m(ery)h(in\014nite)d (sub-)568 990 y(set)j(of)f Fo(X)40 b Fq(is)32 b(connected.)364 1312 y(\(vi\))49 b(\()p Fo(X)r(;)17 b Fm(L)p Fq(\))23 b(is)g(connected)j(except)f(when)g Fo(X)32 b Fq(is)23 b(coun)m(table;)k(indeed,)f(ev)m(ery)g(uncoun)m(t-)568 1433 y(able)32 b(subset)i(of)e Fo(X)40 b Fq(is)32 b(connected.)337 1754 y(\(vii\))48 b(In)40 b(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\),)41 b Fo(A)f Fm(\022)g Fo(R)g Fq(is)f(connected)j(i\013)c Fo(A)i Fq(is)f(an)g(in)m(terv)-5 b(al.)63 b(\(Th)m(us,)43 b(subspaces)568 1875 y(of)34 b(connected)j(spaces)g(are)e Fp(not)g Fq(usually)g(connected)h(|)f(examples)g(ab)s(ound)g(in)568 1995 y(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\).\))324 2283 y Fl(2.3.2)136 b(Characterizations)47 b(of)e(Connectedness)324 2468 y Fk(Lemma)37 b(2.8)49 b Fm(;)27 b(\032)i Fo(A)e Fm(\022)i Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))33 b Fp(is)g Fq(not)g Fp(c)-5 b(onne)g(cte)g(d)33 b(i\013)f(ther)-5 b(e)34 b(exist)f Fm(T)25 b Fp(-op)-5 b(en)33 b(sets)g Fo(G)p Fp(,)324 2588 y Fo(H)40 b Fp(such)32 b(that)h Fo(A)28 b Fm(\022)g Fo(G)17 b Fm([)g Fo(H)8 b Fp(,)32 b Fo(A)17 b Fm(\\)g Fo(G)28 b Fm(6)p Fq(=)g Fm(;)p Fp(,)k Fo(A)17 b Fm(\\)h Fo(H)35 b Fm(6)p Fq(=)27 b Fm(;)33 b Fp(and)e Fo(A)17 b Fm(\\)h Fo(G)e Fm(\\)i Fo(H)35 b Fq(=)27 b Fm(;)p Fp(.)44 b(\(A)-5 b(gain,)324 2709 y(we)34 b(c)-5 b(an)35 b(r)-5 b(eplac)g(e)34 b('op)-5 b(en)-10 b(')34 b(by)h('close)-5 b(d')33 b(her)-5 b(e.\))324 2930 y Fq(Pro)s(of)p 324 2943 235 4 v 32 w(Exercise.)324 3050 y Fp(Note)25 b Fq(By)g(an)f(in)m(terv)-5 b(al)24 b(in)f Fo(R)q Fq(,)k(w)m(e)e(mean)f(an)m(y)i(subset)g Fo(I)32 b Fq(suc)m(h)26 b(that)e(whenev)m(er)k Fo(a)f(<)h(b)g(<)g(c)324 3170 y Fq(and)46 b(whenev)m(er)i Fo(a)i Fm(2)h Fo(I)i Fq(and)46 b Fo(c)k Fm(2)h Fo(I)j Fq(then)46 b Fo(b)51 b Fm(2)f Fo(I)8 b Fq(.)83 b(It)46 b(is)f(routine)g(to)h(c)m(hec)m(k)i (that)324 3291 y(the)35 b(only)g(ones)g(are)g(\()p Fo(a;)17 b(b)p Fq(\),)36 b([)p Fo(a;)17 b(b)p Fq(],)36 b([)p Fo(a;)17 b(b)p Fq(\),)36 b(\()p Fo(a;)17 b(b)p Fq(],)36 b([)p Fo(a;)17 b Fm(1)p Fq(\),)35 b(\()p Fo(a;)17 b Fm(1)p Fq(\),)35 b(\()p Fm(\0001)p Fo(;)17 b(b)p Fq(\),)36 b(\()p Fm(\0001)p Fo(;)17 b(b)p Fq(],)324 3411 y(\()p Fm(\0001)p Fo(;)g Fm(1)p Fq(\))27 b(=)g Fo(R)34 b Fq(and)f Fm(f)p Fo(a)p Fm(g)f Fq(for)g(real)g Fo(a)p Fq(,)g Fo(b)p Fq(,)i Fo(a)27 b(<)h(b)33 b Fq(where)h(appropriate.)324 3531 y(It)e(turns)i(out)e(that)g(these)i(are)f(exactly)g(the)g(connected)h (subsets)h(of)d(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\):-)324 3729 y Fk(Lemma)37 b(2.9)49 b Fp(In)26 b Fo(R)q Fp(,)j(if)e Fq([)p Fo(a;)17 b(b)p Fq(])28 b(=)g Fo(F)1638 3744 y Fh(1)1682 3729 y Fm([)5 b Fo(F)1816 3744 y Fh(2)1883 3729 y Fp(wher)-5 b(e)27 b Fo(F)2214 3744 y Fh(1)2253 3729 y Fp(,)i Fo(F)2375 3744 y Fh(2)2442 3729 y Fp(ar)-5 b(e)26 b(b)-5 b(oth)27 b(close)-5 b(d)27 b(and)f Fo(a)i Fm(2)g Fo(F)3492 3744 y Fh(1)3532 3729 y Fp(,)324 3850 y Fo(b)g Fm(2)g Fo(F)550 3865 y Fh(2)624 3850 y Fp(then)35 b Fo(F)904 3865 y Fh(1)966 3850 y Fm(\\)22 b Fo(F)1117 3865 y Fh(2)1185 3850 y Fm(6)p Fq(=)27 b Fm(;)p Fp(.)324 4047 y Fq(Pro)s(of)p 324 4060 V 32 w(Exercise.)324 4245 y Fk(Theorem)37 b(2.6)49 b Fp(L)-5 b(et)35 b Fm(;)28 b(\032)g Fo(I)35 b Fm(\022)28 b Fq(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\))p Fp(.)45 b(Then)34 b Fo(I)43 b Fp(is)35 b(c)-5 b(onne)g(cte)g(d)34 b(i\013)g Fo(I)43 b Fp(is)34 b(an)h(interval.)324 4443 y Fq(Pro)s(of)p 324 4456 V 392 4641 a Fm(\))p Fq(:)49 b(If)31 b Fo(I)38 b Fq(is)31 b Fp(not)40 b Fq(an)31 b(in)m(terv)-5 b(al,)30 b(then)i(there)f(exist)h Fo(a)c(<)f(b)h(<)g(c)j Fq(with)f Fo(a)e Fm(2)g Fo(I)8 b Fq(,)31 b Fo(b)e Fm(62)f Fo(I)38 b Fq(and)568 4761 y Fo(c)d Fm(2)g Fo(I)8 b Fq(.)56 b(T)-8 b(ak)m(e)38 b Fo(A)d Fq(=)f Fo(I)f Fm(\\)26 b Fq(\()p Fm(\0001)p Fo(;)17 b(b)p Fq(\))37 b(and)g Fo(B)j Fq(=)34 b Fo(I)f Fm(\\)26 b Fq(\()p Fo(b;)17 b Fm(1)p Fq(\).)55 b(Then)38 b Fo(A)26 b Fm([)f Fo(B)40 b Fq(=)35 b Fo(I)8 b Fq(,)568 4882 y Fo(A)22 b Fm(\\)h Fo(B)34 b Fq(=)28 b Fm(;)p Fq(,)34 b Fo(A)28 b Fm(6)p Fq(=)h Fm(;)p Fq(,)k Fo(B)h Fm(6)p Fq(=)28 b Fm(;)p Fq(,)33 b Fo(A)c Fm(\032)g Fo(I)8 b Fq(,)33 b Fo(B)h Fm(\032)29 b Fo(I)41 b Fq(and)33 b Fo(A)p Fq(,)h Fo(B)k Fq(are)33 b(b)s(oth)g(op)s(en)g(in)g Fo(I)568 5002 y Fq(i.e.)43 b Fo(A)32 b Fq(and)h Fo(B)38 b Fq(partition)30 b Fo(I)40 b Fq(and)33 b(so)g Fo(I)40 b Fq(is)32 b(not)h(connected.)1894 5251 y(22)p eop %%Page: 23 24 23 23 bop 392 548 a Fm(\()p Fq(:)49 b(Supp)s(ose)37 b Fo(I)43 b Fq(is)36 b(not)g(connected)h(and)f(that)g Fo(I)44 b Fp(is)f Fq(an)36 b(in)m(terv)-5 b(al.)52 b(By)37 b(the)f(`closed')568 668 y(v)m(ersion)46 b(of)g(Lemma)f(2.8,)k(there)e(exist)g(closed)f (subsets)i Fo(K)2843 683 y Fh(1)2883 668 y Fq(,)i Fo(K)3043 683 y Fh(2)3128 668 y Fq(of)c Fo(R)h Fq(suc)m(h)568 789 y(that)42 b Fo(I)51 b Fm(\022)45 b Fo(K)1088 804 y Fh(1)1156 789 y Fm([)29 b Fo(K)1334 804 y Fh(2)1373 789 y Fq(,)45 b Fo(I)36 b Fm(\\)29 b Fo(K)1702 804 y Fh(1)1785 789 y Fm(6)p Fq(=)44 b Fm(;)p Fq(,)h Fo(I)36 b Fm(\\)29 b Fo(K)2284 804 y Fh(2)2367 789 y Fm(6)p Fq(=)44 b Fm(;)e Fq(and)g Fo(I)37 b Fm(\\)29 b Fo(K)3036 804 y Fh(1)3104 789 y Fm(\\)g Fo(K)3282 804 y Fh(2)3365 789 y Fq(=)44 b Fm(;)p Fq(.)568 909 y(Select)j Fo(a)53 b Fm(2)h Fo(I)40 b Fm(\\)32 b Fo(K)1352 924 y Fh(1)1392 909 y Fq(,)51 b Fo(b)i Fm(2)g Fo(I)40 b Fm(\\)33 b Fo(K)1948 924 y Fh(2)1987 909 y Fq(;)55 b(without)47 b(loss)g(of)g(generalit)m(y)-8 b(,)50 b Fo(a)j(<)g(b)p Fq(.)568 1029 y(Then)37 b([)p Fo(a;)17 b(b)p Fq(])35 b Fm(\022)g Fo(I)44 b Fq(so)36 b(that)h([)p Fo(a;)17 b(b)p Fq(])34 b(=)h(\([)p Fo(a;)17 b(b)p Fq(])25 b Fm(\\)g Fo(K)2351 1044 y Fh(1)2390 1029 y Fq(\))g Fm([)g Fq(\([)p Fo(a;)17 b(b)p Fq(])25 b Fm(\\)h Fo(K)2972 1044 y Fh(2)3011 1029 y Fq(\),)37 b(whence)i(b)m(y)568 1150 y(Lemma)31 b(2.9,)h Fm(;)c(6)p Fq(=)f([)p Fo(a;)17 b(b)p Fq(])23 b Fm(\\)f Fo(K)1665 1165 y Fh(1)1727 1150 y Fm(\\)h Fo(K)1899 1165 y Fh(2)1966 1150 y Fm(\022)28 b Fo(I)i Fm(\\)22 b Fo(K)2315 1165 y Fh(1)2377 1150 y Fm(\\)g Fo(K)2548 1165 y Fh(2)2616 1150 y Fq(=)27 b Fm(;)p Fq(!)324 1438 y Fl(2.3.3)136 b(Connectedness)45 b(and)g(Con)l(tin)l (uous)g(Maps)324 1623 y Fk(Lemma)37 b(2.10)49 b Fp(Conne)-5 b(cte)g(dness)33 b(is)i(pr)-5 b(eserve)g(d)34 b(by)h(c)-5 b(ontinuous)34 b(maps.)324 1850 y Fq(Pro)s(of)p 324 1863 235 4 v 32 w(Exercise.)324 2077 y Fk(Corollary)i(2.2)h(\(In)m (termediate)f(V)-9 b(alue)36 b(Theorem\))48 b Fp(If)37 b Fo(f)43 b Fq(:)32 b([)p Fo(a;)17 b(b)p Fq(])33 b Fm(!)f Fo(R)38 b Fp(is)f(c)-5 b(on-)324 2197 y(tinuous)35 b(and)f Fo(f)11 b Fq(\()p Fo(a)p Fq(\))28 b Fo(<)f(y)k(<)d(f)11 b Fq(\()p Fo(b)p Fq(\))p Fp(,)34 b(then)h Fo(y)j Fp(must)d(b)-5 b(e)35 b(a)g(value)f(of)h Fo(f)11 b Fp(.)324 2424 y Fq(Pro)s(of)p 324 2437 V 32 w(Exercise.)324 2651 y Fk(Corollary)36 b(2.3)h(\(Fixed)g(p)s(oin)m(t)g(theorem)g(for)g Fq([0)p Fo(;)17 b Fq(1])p Fk(\))48 b Fp(If)f Fo(f)61 b Fq(:)50 b([0)p Fo(;)17 b Fq(1])50 b Fm(!)g Fq([0)p Fo(;)17 b Fq(1])47 b Fp(is)324 2771 y(c)-5 b(ontinuous,)40 b(then)f(it)h(has)f(a) g(`\014xe)-5 b(d)39 b(p)-5 b(oint')39 b(i.e.)58 b(ther)-5 b(e)39 b(exists)g(some)g Fo(x)d Fm(2)h Fq([0)p Fo(;)17 b Fq(1])39 b Fp(such)324 2891 y(that)c Fo(f)11 b Fq(\()p Fo(x)p Fq(\))28 b(=)f Fo(x)p Fp(.)324 3118 y Fq(Pro)s(of)p 324 3131 V 31 w(Consider)33 b Fo(g)t Fq(\()p Fo(x)p Fq(\))27 b(=)h Fo(f)11 b Fq(\()p Fo(x)p Fq(\))21 b Fm(\000)g Fo(x)p Fq(.)44 b(Then)33 b Fo(g)e Fq(:)d([0)p Fo(;)17 b Fq(1])27 b Fm(!)h Fo(R)33 b Fq(is)e(con)m(tin)m(uous.)44 b(F)-8 b(urther,)324 3238 y Fo(g)t Fq(\(0\))30 b(=)h Fo(f)11 b Fq(\(0\))31 b Fm(\025)h Fq(0)i(and)h Fo(g)t Fq(\(1\))30 b(=)i Fo(f)11 b Fq(\(1\))22 b Fm(\000)j Fq(1)31 b Fm(\024)g Fq(0)k(so)g(that)f(0)h(is)f(in)m(termediate)g(b)s(et)m(w)m(een)324 3359 y Fo(g)t Fq(\(0\))i(and)g Fo(g)t Fq(\(1\).)56 b(Th)m(us,)39 b(b)m(y)f(the)f(In)m(termediate)g(V)-8 b(alue)36 b(Theorem,)j(there)e (exists)h Fo(x)d Fm(2)324 3479 y Fq([0)p Fo(;)17 b Fq(1])32 b(suc)m(h)i(that)e(0)c(=)f Fo(g)t Fq(\()p Fo(x)p Fq(\))g(=)h Fo(f)11 b Fq(\()p Fo(x)p Fq(\))22 b Fm(\000)h Fo(x)33 b Fq(i.e.)43 b(suc)m(h)34 b(that)e Fo(f)11 b Fq(\()p Fo(x)p Fq(\))28 b(=)f Fo(x)p Fq(.)324 3600 y Fp(Note)37 b Fq(Giv)m(en)f(con)m(tin)m(uous)h Fo(h)d Fq(:)h([)p Fo(a;)17 b(b)p Fq(])35 b Fm(!)f Fq([)p Fo(a;)17 b(b)p Fq(],)38 b(it)d(follo)m(ws)g(that)h Fo(h)h Fq(has)g(a)f(\014xed)h(p)s (oin)m(t)324 3720 y(since)46 b([)p Fo(a;)17 b(b)p Fq(])815 3692 y Fm(\030)816 3724 y Fq(=)942 3720 y([0)p Fo(;)g Fq(1])45 b(and)g(`ev)m(ery)j(con)m(tin)m(uous)e(function)e(has)i(a)f (\014xed)i(p)s(oin)m(t')d(is)h(a)324 3840 y(homeomorphic)31 b(in)m(v)-5 b(arian)m(t.)324 4067 y Fk(Lemma)37 b(2.11)49 b Fp(L)-5 b(et)37 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))37 b Fp(b)-5 b(e)37 b(disc)-5 b(onne)g(cte)g(d)35 b(with)i Fm(;)32 b(\032)g Fo(Y)53 b Fm(\032)32 b Fo(X)8 b Fp(,)37 b Fo(Y)59 b Fp(clop)-5 b(en.)50 b(If)37 b Fo(A)324 4187 y Fp(is)d(any)h(c)-5 b(onne)g(cte)g(d)34 b(subset)h(of)g Fo(X)8 b Fp(,)34 b(then)h Fo(A)28 b Fm(\022)g Fo(Y)56 b Fp(or)35 b Fo(A)27 b Fm(\022)i Fo(X)h Fm(n)22 b Fo(Y)f Fp(.)324 4414 y Fq(Pro)s(of)p 324 4427 V 41 w(If)42 b Fo(A)29 b Fm(\\)g Fo(Y)65 b Fm(6)p Fq(=)44 b Fm(;)g(6)p Fq(=)f Fo(A)29 b Fm(\\)g Fq(\()p Fo(X)36 b Fm(n)29 b Fo(Y)21 b Fq(\),)45 b(then)d Fm(;)i(\032)g Fo(A)29 b Fm(\\)g Fo(Y)65 b Fm(\032)45 b Fo(A)d Fq(and)g Fo(A)29 b Fm(\\)g Fo(Y)63 b Fq(is)324 4535 y(clop)s(en)34 b Fp(in)j Fo(A)p Fq(.)51 b(Th)m(us,)37 b Fo(A)e Fq(is)f Fp(not)h Fq(connected!)53 b(It)35 b(follo)m(ws)e(that)i(either)g Fo(A)24 b Fm(\\)g Fo(Y)53 b Fq(=)31 b Fm(;)k Fq(or)324 4655 y Fo(A)22 b Fm(\\)h Fo(X)30 b Fm(n)21 b Fo(Y)49 b Fq(=)28 b Fm(;)k Fq(i.e.)43 b(either)33 b Fo(A)28 b Fm(\022)g Fo(X)i Fm(n)22 b Fo(Y)53 b Fq(or)33 b Fo(A)27 b Fm(\022)i Fo(Y)21 b Fq(.)324 4882 y Fk(Lemma)37 b(2.12)49 b Fp(If)e(the)g(family)g Fm(f)p Fo(A)1690 4897 y Fj(i)1769 4882 y Fq(:)k Fo(i)g Fm(2)g Fo(I)8 b Fm(g)47 b Fp(of)h(c)-5 b(onne)g(cte)g(d)46 b(subsets)h(of)g(a)g(sp)-5 b(ac)g(e)324 5002 y Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))35 b Fp(has)g(a)f (non-empty)g(interse)-5 b(ction,)35 b(then)f(its)h(union)g Fm([)2639 5017 y Fj(i)p Fg(2)p Fj(I)2750 5002 y Fo(A)2823 5017 y Fj(i)2886 5002 y Fp(is)g(c)-5 b(onne)g(cte)g(d.)1894 5251 y Fq(23)p eop %%Page: 24 25 24 24 bop 324 548 a Fq(Pro)s(of)p 324 561 235 4 v 26 w(Supp)s(ose)28 b(not)e(and)h(that)g(there)h(exists)f(a)g(non-empt)m(y) g(prop)s(er)g(clop)s(en)f(subset)i Fo(Y)324 668 y Fq(of)33 b Fm([)502 683 y Fj(i)p Fg(2)p Fj(I)614 668 y Fo(A)687 683 y Fj(i)715 668 y Fq(.)47 b(Then)35 b(for)e(eac)m(h)i Fo(i)30 b Fm(2)g Fo(I)8 b Fq(,)34 b(either)g Fo(A)2038 683 y Fj(i)2096 668 y Fm(\022)c Fo(Y)55 b Fq(or)34 b Fo(A)2509 683 y Fj(i)2567 668 y Fm(\022)c([)2740 683 y Fj(i)p Fg(2)p Fj(I)2852 668 y Fo(A)2925 683 y Fj(i)2976 668 y Fm(n)23 b Fo(Y)e Fq(.)47 b(Ho)m(w)m(ev)m(er)324 789 y(if)33 b(for)g(some)h Fo(j)6 b Fq(,)35 b Fo(A)992 804 y Fj(j)1059 789 y Fm(\022)c Fo(Y)21 b Fq(,)35 b(then)f Fo(A)1603 804 y Fj(i)1662 789 y Fm(\022)d Fo(Y)55 b Fq(for)34 b(eac)m(h)h Fo(i)c Fm(2)f Fo(I)42 b Fq(\(since)35 b Fm(\\)2844 804 y Fj(i)p Fg(2)p Fj(I)2955 789 y Fo(A)3028 804 y Fj(i)3087 789 y Fm(6)p Fq(=)30 b Fm(;)p Fq(\))k(whic)m(h)324 909 y(implies)c(that)i Fm([)932 924 y Fj(i)p Fg(2)p Fj(I)1044 909 y Fo(A)1117 924 y Fj(i)1173 909 y Fm(\022)c Fo(Y)21 b Fq(!)324 1029 y(Similarly)-8 b(,)28 b(if)k(for)g(some)g Fo(k)f Fm(2)d Fo(I)8 b Fq(,)33 b Fo(A)1600 1044 y Fj(k)1670 1029 y Fm(\022)28 b([)1841 1044 y Fj(i)p Fg(2)p Fj(I)1953 1029 y Fo(A)2026 1044 y Fj(i)2076 1029 y Fm(n)22 b Fo(Y)f Fq(,)33 b(then)g Fm([)2574 1044 y Fj(i)p Fg(2)p Fj(I)2685 1029 y Fo(A)2758 1044 y Fj(i)2814 1029 y Fm(\022)c([)2986 1044 y Fj(i)p Fg(2)p Fj(I)3097 1029 y Fo(A)3170 1044 y Fj(i)3220 1029 y Fm(n)22 b Fo(Y)g Fq(!)324 1254 y Fk(Corollary)36 b(2.4)49 b Fp(Given)38 b(a)g(family)g Fm(f)p Fo(C)1810 1269 y Fj(i)1872 1254 y Fq(:)d Fo(i)f Fm(2)h Fo(I)8 b Fm(g)38 b Fp(of)g(c)-5 b(onne)g(cte)g(d)38 b(subsets)g(of)g(a)g(sp)-5 b(ac)g(e)324 1375 y Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))p Fp(,)49 b(if)d Fo(B)54 b Fm(\022)c Fo(X)k Fp(is)46 b(also)f(c)-5 b(onne)g(cte)g(d)46 b(and)f Fo(B)36 b Fm(\\)31 b Fo(C)2437 1390 y Fj(i)2514 1375 y Fm(6)p Fq(=)49 b Fm(;)d Fp(for)g(al)5 b(l)46 b Fo(i)j Fm(2)g Fo(I)8 b Fp(,)49 b(then)324 1495 y Fo(B)27 b Fm([)c Fq(\()p Fm([)618 1510 y Fj(i)p Fg(2)p Fj(I)729 1495 y Fo(C)799 1510 y Fj(i)827 1495 y Fq(\))35 b Fp(is)f(c)-5 b(onne)g(cte)g(d.)324 1720 y Fq(Pro)s(of)p 324 1733 V 32 w(T)d(ak)m(e)33 b Fo(A)899 1735 y Fj(i)955 1720 y Fq(=)28 b Fo(B)f Fm([)c Fo(C)1319 1735 y Fj(i)1379 1720 y Fq(in)32 b(Lemma)f(2.12.)324 1944 y Fk(Lemma)37 b(2.13)49 b Fp(If)27 b Fo(A)h Fp(is)g(a)f(c)-5 b(onne)g(cte)g(d)27 b(subset)h(of)f(a)h(sp)-5 b(ac)g(e)27 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))j Fp(and)f Fo(A)h Fm(\022)g Fo(B)33 b Fm(\022)3424 1919 y Fq(\026)3398 1944 y Fo(A)3471 1908 y Fg(T)3532 1944 y Fp(,)324 2065 y(then)h Fo(B)40 b Fp(is)35 b(a)g(c)-5 b(onne)g(cte)g(d)34 b(subset.)324 2289 y Fq(Pro)s(of)p 324 2302 V 37 w(If)k Fo(B)43 b Fq(is)38 b Fp(not)48 b Fq(connected,)41 b(then)d(there)h(exists)g Fm(;)e(\032)h Fo(Y)58 b Fm(\032)38 b Fo(B)43 b Fq(whic)m(h)38 b(is)g(clop)s(en)324 2410 y Fp(in)j Fo(B)5 b Fq(.)67 b(By)41 b(Lemma)e(2.11,)j(either)e Fo(A)h Fm(\022)h Fo(Y)61 b Fq(or)40 b Fo(A)h Fm(\022)h Fo(B)32 b Fm(n)27 b Fo(Y)22 b Fq(.)67 b(Supp)s(ose)41 b Fo(A)g Fm(\022)g Fo(Y)62 b Fq(\(a)324 2530 y(similar)35 b(argumen)m(t)j(su\016ces)j(for)c Fo(A)h Fm(\022)g Fo(B)32 b Fm(n)26 b Fo(Y)21 b Fq(\);)41 b(then)2430 2505 y(\026)2404 2530 y Fo(A)2477 2494 y Fg(T)2575 2530 y Fm(\022)2705 2505 y Fq(\026)2691 2530 y Fo(Y)2769 2494 y Fg(T)2867 2530 y Fq(and)e(so)g Fo(B)31 b Fm(n)26 b Fo(Y)59 b Fq(=)324 2651 y Fo(B)32 b Fm(\\)c Fq(\()p Fo(B)k Fm(n)27 b Fo(Y)21 b Fq(\))41 b(=)1044 2625 y(\026)1018 2651 y Fo(A)1091 2614 y Fg(T)1130 2625 y Fi(B)1214 2651 y Fm(\\)28 b Fq(\()p Fo(B)k Fm(n)27 b Fo(Y)21 b Fq(\))40 b Fm(\022)1818 2625 y Fq(\026)1803 2651 y Fo(Y)1882 2614 y Fg(T)1921 2625 y Fi(B)2005 2651 y Fm(\\)27 b Fq(\()p Fo(B)33 b Fm(n)27 b Fo(Y)21 b Fq(\))40 b(=)g Fo(Y)49 b Fm(\\)27 b Fq(\()p Fo(B)33 b Fm(n)27 b Fo(Y)21 b Fq(\))40 b(=)h Fm(;)e Fq(|)h(a)324 2771 y(con)m(tradiction!)324 2996 y Fk(De\014nition)c(2.7)49 b Fp(L)-5 b(et)36 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))36 b Fp(b)-5 b(e)35 b(a)h(top)-5 b(olo)g(gic)g(al)34 b(sp)-5 b(ac)g(e)35 b(with)g Fo(x)30 b Fm(2)f Fo(X)8 b Fp(;)36 b(we)f(de\014ne)g(the)324 3116 y Fk(comp)s(onen)m(t)k(of)g Fo(x)p Fp(,)e Fo(C)1221 3131 y Fj(x)1265 3116 y Fp(,)g(in)f Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))36 b Fp(to)h(b)-5 b(e)36 b(the)g(union)g(of)g(al)5 b(l)36 b(c)-5 b(onne)g(cte)g(d)36 b(subsets)g(of)324 3236 y Fo(X)42 b Fp(which)34 b(c)-5 b(ontain)35 b Fo(x)g Fp(i.e.)989 3453 y Fo(C)1059 3468 y Fj(x)1130 3453 y Fq(=)27 b Fm([f)p Fo(A)h Fm(\022)h Fo(X)35 b Fq(:)28 b Fo(x)g Fm(2)g Fo(A)35 b Fp(and)f Fo(A)h Fp(is)g(c)-5 b(onne)g(cte)g(d)o Fm(g)p Fo(:)324 3678 y Fq(F)d(or)35 b(eac)m(h)i Fo(x)d Fm(2)g Fo(X)8 b Fq(,)37 b(it)e(follo)m(ws)f(from)h(Lemma)g(2.12)g(that)h Fo(C)2571 3693 y Fj(x)2650 3678 y Fq(is)g(the)g Fp(maximum)43 b Fq(con-)324 3798 y(nected)32 b(subset)h(of)e Fo(X)39 b Fq(whic)m(h)31 b(con)m(tains)g Fo(x)p Fq(.)44 b(Also)30 b(it)h(is)f(clear)h(that)g(if)f Fo(x)p Fq(,)i Fo(y)e Fm(2)f Fo(X)8 b Fq(,)31 b(either)324 3919 y Fo(C)394 3934 y Fj(x)469 3919 y Fq(=)h Fo(C)647 3934 y Fj(y)723 3919 y Fq(or)i Fo(C)914 3934 y Fj(x)982 3919 y Fm(\\)24 b Fo(C)1142 3934 y Fj(y)1215 3919 y Fq(=)31 b Fm(;)k Fq(\(for)f(if)g Fo(z)i Fm(2)c Fo(C)1937 3934 y Fj(x)2005 3919 y Fm(\\)24 b Fo(C)2165 3934 y Fj(y)2206 3919 y Fq(,)36 b(then)f Fo(C)2563 3934 y Fj(x)2630 3919 y Fm([)25 b Fo(C)2791 3934 y Fj(y)2863 3919 y Fm(\022)33 b Fo(C)3043 3934 y Fj(z)3114 3919 y Fm(\022)f Fo(C)3293 3934 y Fj(x)3360 3919 y Fm(\\)24 b Fo(C)3520 3934 y Fj(y)324 4039 y Fq(whence)36 b Fo(C)737 4054 y Fj(x)812 4039 y Fq(=)30 b Fo(C)988 4054 y Fj(y)1029 4039 y Fq(\(=)h Fo(C)1244 4054 y Fj(z)1284 4039 y Fq(\)\).)49 b(Th)m(us)36 b(w)m(e)f(ma)m(y)f(sp)s(eak)i(of)e Fp(the)41 b Fq(comp)s(onen)m(ts)35 b(of)f(a)g(space)324 4159 y(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))35 b(\(without)g(reference)i (to)d(sp)s(eci\014c)i(p)s(oin)m(ts)f(of)g Fo(X)8 b Fq(\):)48 b(they)36 b(partition)d(the)i(space)324 4280 y(in)m(to)25 b(connected)i Fp(close)-5 b(d)24 b Fq(subsets)k(\(b)m(y)e(Lemma)e (2.13\))h(and)h(are)f(precisely)h(the)g Fp(maximal)324 4400 y Fq(connected)34 b(subsets)h(of)d Fo(X)8 b Fq(.)324 4521 y Fp(Examples)324 4641 y Fq(\(i\))31 b(If)i(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))33 b(is)f(connected,)i(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))32 b(has)h(only)f(one)h(comp)s(onen)m(t,) g(namely)f Fo(X)8 b Fq(!)324 4761 y(\(ii\))30 b(F)-8 b(or)32 b(an)m(y)h(discrete)h(space,)f(the)g(comp)s(onen)m(ts)g(are)g (the)g(singletons.)324 4882 y(\(iii\))20 b(In)j Fo(Q)h Fq(\(with)f(its)f(usual)h(top)s(ology\),)g(the)h(comp)s(onen)m(ts)g (are)f(the)g(singletons.)40 b(\(Th)m(us,)324 5002 y(comp)s(onen)m(ts)33 b(need)g(not)g(b)s(e)g(op)s(en.\))1894 5251 y(24)p eop %%Page: 25 26 25 25 bop 324 548 a Fk(De\014nition)36 b(2.8)49 b Fp(A)40 b(sp)-5 b(ac)g(e)39 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))40 b Fp(is)g Fk(totally)h(disconnected)f Fp(i\013)f(the)h(only)f (c)-5 b(on-)324 668 y(ne)g(cte)g(d)26 b(subsets)h(of)g Fo(X)35 b Fp(ar)-5 b(e)27 b(the)g(singletons)f(\(e)-5 b(quivalently,)28 b(the)g(c)-5 b(omp)g(onents)25 b(of)i Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))324 789 y Fp(ar)-5 b(e)34 b(the)h(singletons\).)324 978 y Fq(Th)m(us,)43 b(b)m(y)e(the)f(previous)g(examples,)h(w)m(e)g(see)g(that)e(the)i (space)f Fo(Q)g Fq(of)g(rationals,)f(the)324 1098 y(space)30 b Fo(R)16 b Fm(n)f Fo(Q)30 b Fq(of)e(irrationals)e(and)k(an)m(y)g (discrete)f(space)i(are)e(all)e(totally)g(disconnected.)324 1218 y(F)-8 b(urther,)32 b(the)h(Sorgenfrey)h(line)d Fo(R)1613 1233 y Fj(s)1683 1218 y Fq(is)h(totally)e(disconnected.)324 1505 y Fl(2.3.4)136 b(P)l(ath)l(wise)46 b(Connectedness)324 1689 y Fk(De\014nition)36 b(2.9)49 b Fp(A)39 b(top)-5 b(olo)g(gic)g(al)39 b(sp)-5 b(ac)g(e)38 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))39 b Fp(is)g Fk(path)m(wise)j(connected)e Fp(i\013)f(for)324 1810 y(any)i Fo(x)p Fp(,)i Fo(y)f Fm(2)e Fo(X)8 b Fp(,)42 b(ther)-5 b(e)41 b(exists)g(a)g(c)-5 b(ontinuous)41 b(function)g Fo(f)50 b Fq(:)39 b([0)p Fo(;)17 b Fq(1])39 b Fm(!)f Fo(X)49 b Fp(such)41 b(that)324 1930 y Fo(f)11 b Fq(\(0\))27 b(=)g Fo(x)36 b Fp(and)e Fo(f)11 b Fq(\(1\))27 b(=)h Fo(y)t Fp(.)43 b(Such)35 b(a)g(function)f Fo(f)46 b Fp(is)34 b(c)-5 b(al)5 b(le)-5 b(d)34 b(a)h Fk(path)h Fp(fr)-5 b(om)34 b Fo(x)h Fp(to)g Fo(y)t Fp(.)324 2139 y Fk(Theorem)i(2.7)49 b Fp(Every)35 b(p)-5 b(athwise)34 b(c)-5 b(onne)g(cte)g(d)34 b(sp)-5 b(ac)g(e)34 b(is)g(c)-5 b(onne)g(cte)g(d.)324 2348 y Fq(Pro)s(of)p 324 2361 235 4 v 22 w(Let)23 b(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))c(b)s(e)h(path)m(wise)h(connected)g(and)f(let)f Fo(a)28 b Fm(2)g Fo(X)8 b Fq(;)26 b(for)c(ev)m(ery)j Fo(x)j Fm(2)g Fo(X)8 b Fq(,)25 b(there)324 2469 y(exists)38 b(a)f(path)g Fo(p)965 2484 y Fj(x)1044 2469 y Fq(:)f([0)p Fo(;)17 b Fq(1])35 b Fm(!)g Fo(X)45 b Fq(from)36 b Fo(a)h Fq(to)g Fo(x)p Fq(.)57 b(Then,)40 b(for)c(eac)m(h)j Fo(x)c Fm(2)h Fo(X)8 b Fq(,)38 b Fo(p)3246 2484 y Fj(x)3290 2469 y Fq(\([0)p Fo(;)17 b Fq(1]\))324 2589 y(is)36 b(connected;)k (moreo)m(v)m(er,)d Fo(p)1414 2604 y Fj(a)1456 2589 y Fq(\(0\))d(=)g Fo(a)g Fm(2)g(\\)1976 2604 y Fj(x)p Fg(2)p Fj(X)2131 2589 y Fo(p)2180 2604 y Fj(x)2224 2589 y Fq(\([0)p Fo(;)17 b Fq(1]\))35 b(so)i(that)f(b)m(y)h(Lemma)e(2.12,)324 2709 y Fo(X)g Fq(=)28 b Fm([)610 2724 y Fj(x)p Fg(2)p Fj(X)764 2709 y Fo(p)813 2724 y Fj(x)857 2709 y Fq(\([0)p Fo(;)17 b Fq(1]\))32 b(is)g(connected.)324 2830 y Fp(Note)k(wel)5 b(l)42 b Fq(The)34 b(con)m(v)m(erse)i(is)d(false.)44 b(Consider)34 b(the)g(follo)m(wing)c(example,)j Fp(the)i(top)-5 b(olo-)324 2950 y(gist's)34 b(sine)h(curve)7 b Fq(:)1059 3192 y Fo(V)49 b Fq(=)28 b Fm(f)p Fq(\()p Fo(x;)17 b Fq(0\))27 b(:)h Fo(x)g Fm(\024)g Fq(0)p Fm(g)22 b([)g(f)p Fq(\()p Fo(x;)17 b Fq(sin)2359 3125 y(1)p 2355 3169 56 4 v 2355 3261 a Fo(x)2421 3192 y Fq(\))28 b(:)f Fo(x)h(>)g Fq(0)p Fm(g)324 3426 y Fq(is)45 b(a)g(connected)i(space,)j(but)45 b(no)h(path)f(can)h(b)s(e)f(found)h(from)e(\(0)p Fo(;)17 b Fq(0\))44 b(to)h(an)m(y)h(p)s(oin)m(t)324 3547 y(\()p Fo(x;)17 b Fq(sin)609 3508 y Fh(1)p 607 3523 40 4 v 607 3581 a Fj(x)657 3547 y Fq(\))32 b(with)h Fo(x)28 b(>)f Fq(0)324 3667 y(\(for)37 b(supp)s(ose,)k(w.l.o.g.,)d(there)h(exists)g (a)e(path)h Fo(p)f Fq(:)g([0)p Fo(;)17 b Fq(1])36 b Fm(!)g Fo(X)46 b Fq(with)38 b Fo(p)p Fq(\(0\))e(=)g(\()3382 3628 y Fh(1)p 3378 3644 43 4 v 3378 3701 a Fj(\031)3431 3667 y Fo(;)17 b Fq(0\))324 3787 y(and)25 b Fo(p)p Fq(\(1\))i(=)h(\(0)p Fo(;)17 b Fq(0\).)40 b(Then)26 b Fo(\031)1398 3802 y Fh(1)1445 3787 y Fm(\016)7 b Fo(p)p Fq(,)26 b(b)s(eing)e(con)m(tin)m (uous,)k(m)m(ust)d(tak)m(e)h(all)d(v)-5 b(alues)25 b(b)s(et)m(w)m(een) 324 3908 y(0)30 b(and)604 3869 y Fh(1)p 600 3885 V 600 3942 a Fj(\031)653 3908 y Fq(,)h(in)e(particular)1403 3869 y Fh(1)p 1280 3885 282 4 v 1280 3947 a(\(2)p Fj(n)p Fh(+)1450 3920 y Fe(1)p 1450 3932 31 4 v 1450 3973 a(2)1491 3947 y Fh(\))p Fj(\031)1601 3908 y Fq(for)h(eac)m(h)h Fo(n)g Fq(i.e.)42 b(there)31 b(exists)g Fo(t)2770 3923 y Fj(n)2845 3908 y Fm(2)d Fq([0)p Fo(;)17 b Fq(1])30 b(suc)m(h)i(that)324 4064 y Fo(\031)379 4079 y Fh(1)427 4064 y Fm(\016)9 b Fo(p)p Fq(\()p Fo(t)608 4079 y Fj(n)655 4064 y Fq(\))28 b(=)957 4024 y Fh(1)p 834 4040 282 4 v 834 4103 a(\(2)p Fj(n)p Fh(+)1004 4076 y Fe(1)p 1005 4088 31 4 v 1005 4129 a(2)1045 4103 y Fh(\))p Fj(\031)1152 4064 y Fq(for)d(eac)m(h)i Fo(n)p Fq(.)42 b(Th)m(us,)29 b Fo(p)p Fq(\()p Fo(t)2025 4079 y Fj(n)2072 4064 y Fq(\))e(=)h(\()2412 4024 y Fh(1)p 2289 4040 282 4 v 2289 4103 a(\(2)p Fj(n)p Fh(+)2459 4076 y Fe(1)p 2459 4088 31 4 v 2459 4129 a(2)2500 4103 y Fh(\))p Fj(\031)2580 4064 y Fo(;)17 b Fq(1\))27 b Fm(!)g Fq(\(0)p Fo(;)17 b Fq(1\))25 b(as)i Fo(n)h Fm(!)f(1)p Fq(.)324 4212 y(No)m(w)36 b Fo(t)584 4227 y Fj(n)663 4212 y Fm(2)c Fq([0)p Fo(;)17 b Fq(1])35 b(for)f(all)f Fo(n)j Fq(whic)m(h)f(implies)e(that)i(there)h(exists)g(a)f(subsequence) j(\()p Fo(t)3438 4227 y Fj(n)3481 4239 y Fi(k)3524 4212 y Fq(\))324 4332 y(in)h([0)p Fo(;)17 b Fq(1])38 b(with)h Fo(t)943 4347 y Fj(n)986 4359 y Fi(k)1068 4332 y Fm(!)g Fo(\025)p Fq(.)64 b(Then)41 b Fo(p)p Fq(\()p Fo(t)1739 4347 y Fj(n)1782 4359 y Fi(k)1824 4332 y Fq(\))f Fm(!)f Fo(p)p Fq(\()p Fo(\025)p Fq(\))g(and)g(so)h Fo(\031)2640 4347 y Fh(1)2707 4332 y Fm(\016)26 b Fo(p)p Fq(\()p Fo(t)2905 4347 y Fj(n)2948 4359 y Fi(k)2991 4332 y Fq(\))39 b Fm(!)g Fq(0.)64 b(Th)m(us)324 4453 y Fo(p)p Fq(\()p Fo(\025)p Fq(\))27 b(=)h(\(0)p Fo(;)17 b(y)t Fq(\))31 b(for)h(some)g Fo(y)t Fq(,)g(whence)i Fo(y)d Fq(=)d(0)k(\(since)h Fo(p)p Fq(\()p Fo(\025)p Fq(\))27 b Fm(2)h Fo(X)8 b Fq(\)!\))324 4783 y Fn(2.4)160 b(Separabilit)l(y)324 5002 y Fk(De\014nition)36 b(2.10)49 b Fp(A)35 b(top)-5 b(olo)g(gic)g(al)34 b(sp)-5 b(ac)g(e)34 b(is)h(said)f(to)h(b)-5 b(e)1894 5251 y Fq(25)p eop %%Page: 26 27 26 26 bop 409 548 a Fp(\(i\))49 b Fk(separable)35 b Fp(i\013)g(it)g (has)f(a)h(c)-5 b(ountable)35 b(dense)f(subset.)379 751 y(\(ii\))49 b Fk(completely)39 b(separable)f Fp(\(e)-5 b(quivalently,)p Fk(second)41 b(coun)m(table)p Fp(\))d(i\013)f(it)i (has)e(a)568 872 y(c)-5 b(ountable)34 b(b)-5 b(ase.)324 1100 y(Examples)416 1303 y Fq(\(i\))48 b(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\))32 b(is)h(separable)f(\(since)1682 1278 y(\026)1660 1303 y Fo(Q)c Fq(=)g Fo(R)q Fq(\).)389 1507 y(\(ii\))47 b(\()p Fo(X)r(;)17 b Fm(C)6 b Fq(\))32 b(is)g(separable)h (for)f(an)m(y)h Fo(X)8 b Fq(.)362 1710 y(\(iii\))46 b(\()p Fo(R)q(;)17 b Fm(L)p Fq(\))32 b(is)g Fp(not)g Fq(separable.)324 1939 y Fk(Theorem)37 b(2.8)134 b Fp(\(i\))49 b(Complete)34 b(sep)-5 b(ar)g(ability)34 b(implies)g(sep)-5 b(ar)g(ability.)379 2142 y(\(ii\))49 b(The)34 b(c)-5 b(onverse)34 b(is)g(true)i(in)e (metric)h(sp)-5 b(ac)g(es.)324 2370 y Fq(Pro)s(of)p 324 2383 235 4 v 40 w(W)d(e)40 b(pro)m(v)m(e)i(only)e(\(ii\).)65 b(In)41 b(metric)e(space)i(\()p Fo(M)5 b(;)17 b(d)p Fq(\),)42 b(let)e Fo(D)k Fq(=)d Fm(f)p Fo(x)3035 2385 y Fh(1)3074 2370 y Fo(;)17 b(x)3173 2385 y Fh(2)3213 2370 y Fo(;)g(:)g(:)g(:)o Fm(g)40 b Fq(b)s(e)324 2491 y(dense.)51 b(Consider)36 b Fm(B)e Fq(=)e Fm(f)p Fo(B)5 b Fq(\()p Fo(x)1473 2506 y Fj(i)1501 2491 y Fo(;)17 b(q)t Fq(\))31 b(:)g Fo(i)h Fm(2)g Fo(!)t(;)17 b(q)34 b Fm(2)e Fo(Q;)17 b(q)35 b(>)c Fq(0)p Fm(g)p Fq(,)k(a)g(coun)m(table)g(collection)324 2611 y(of)d(op)s(en)h(sets.)44 b(One)33 b(can)g(sho)m(w)h(that)e Fm(B)k Fq(is)c(a)h(base)g(for)f Fm(T)2438 2626 y Fj(d)2511 2611 y Fq(.)16 b(.)g(.)g(o)m(v)m(er)35 b(to)d(y)m(ou!)324 2814 y Fk(Theorem)37 b(2.9)134 b Fp(\(i\))49 b(Complete)34 b(sep)-5 b(ar)g(ability)34 b(is)h(her)-5 b(e)g(ditary.)379 3018 y(\(ii\))49 b(Sep)-5 b(ar)g(ability)39 b(is)i(not)f(her)-5 b(e)g(ditary.)61 b(\(Consider)39 b(the)h(`include)-5 b(d)40 b(p)-5 b(oint')40 b(top)-5 b(olo)g(gy)568 3138 y Fm(I)7 b Fq(\(0\))27 b Fp(on)g Fo(R)q Fp(;)i(then)e Fq(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\(0\)\))27 b(is)g Fp(sep)-5 b(ar)g(able,)27 b(sinc)-5 b(e)p 2431 3054 149 4 v 26 w Fm(f)p Fq(0)p Fm(g)27 b Fq(=)h Fo(R)q Fp(.)42 b(However,)28 b Fo(R)6 b Fm(n)f(f)p Fq(0)p Fm(g)568 3259 y Fp(is)34 b Fq(not)h Fp(sep)-5 b(ar)g(able)34 b(b)-5 b(e)g(c)g(ause)34 b(it)h(is)g(discr)-5 b(ete.\))324 3462 y(Example)324 3582 y Fq(Separabilit)m(y)20 b(do)s(es)i Fp(not)31 b Fq(imply)20 b(complete)h(separabilit)m(y)f(since,)25 b(for)c(example,)j(\()p Fo(R)q(;)17 b Fm(I)7 b Fq(\(0\)\))324 3703 y(is)33 b(separable)h(but)g Fp(not)43 b Fq(completely)32 b(separable.\(Supp)s(ose)j(there)g(exists)f(a)f(coun)m(table)324 3823 y(base)i Fm(B)k Fq(for)34 b(its)h(top)s(ology)-8 b(.)48 b(Giv)m(en)35 b Fo(x)d Fm(6)p Fq(=)g(0,)j Fm(f)p Fq(0)p Fo(;)17 b(x)p Fm(g)35 b Fq(is)f(an)h(op)s(en)g(neigh)m(b)s (ourho)s(o)s(d)f(of)g Fo(x)324 3944 y Fq(and)g(so)g(there)h(exists)g Fo(B)1232 3959 y Fj(x)1306 3944 y Fm(2)c(B)37 b Fq(suc)m(h)f(that)e Fo(x)c Fm(2)h Fo(B)2196 3959 y Fj(x)2270 3944 y Fm(\022)g(f)p Fq(0)p Fo(;)17 b(x)p Fm(g)p Fq(.Th)m(us)35 b Fo(B)2975 3959 y Fj(x)3049 3944 y Fq(=)30 b Fm(f)p Fq(0)p Fo(;)17 b(x)p Fm(g)34 b Fq(i.e.)324 4064 y Fm(B)i Fq(is)c(uncoun)m(table)h(.)16 b(.)g(.)g(con)m(tradiction!)324 4292 y Fk(Theorem)37 b(2.10)49 b Fp(Sep)-5 b(ar)g(ability)34 b(is)h(pr)-5 b(eserve)g(d)34 b(by)h(c)-5 b(ontinuous)34 b(maps.)324 4521 y Fq(Pro)s(of)p 324 4534 235 4 v 32 w(Exercise.)324 4641 y Fp(Note)40 b Fq(Complete)32 b(separabilit)m(y)f(is)h Fp(not)42 b Fq(preserv)m(ed)36 b(b)m(y)d(con)m(tin)m(uous)g(maps.)1894 5251 y(26)p eop %%Page: 27 28 27 27 bop 324 1212 a Fr(Chapter)78 b(3)324 1627 y(Con)-6 b(v)g(ergence)324 2080 y Fq(In)30 b(Chapter)h(1,)g(w)m(e)g(de\014ned)h (limits)27 b(of)i(sequences)34 b(in)29 b(a)h(top)s(ological)c(space)32 b(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))30 b(so)324 2200 y(as)23 b(to)g(assimilate)d(the)k(metric)e(de\014nition.)39 b(W)-8 b(e)24 b(noted,)h(ho)m(w)m(ev)m(er,)j(that)23 b(not)g(ev)m(erything)324 2320 y(w)m(e)33 b(knew)h(ab)s(out)f(this)f (idea)g(in)f(metric)h(spaces)i(is)e(v)-5 b(alid)31 b(in)h(top)s (ological)d(spaces.)324 2441 y(W)-8 b(e)33 b(will)d(examine)i(t)m(w)m (o)h(main)e(w)m(a)m(ys)k(around)d(this)g(di\016cult)m(y:)469 2644 y Fm(\017)49 b Fq(dev)m(elop)31 b(a)f(kind)h(of)f(`sup)s (er-sequence')k(or)c Fp(net)40 b Fq(whic)m(h)31 b(do)s(es)g(for)g (general)f(top)s(ol-)568 2765 y(ogy)i(what)h(ordinary)f(sequences)k(do) c(for)g(metric)g(spaces.)469 2968 y Fm(\017)49 b Fq(iden)m(tify)21 b(the)h(class)g(of)g(top)s(ological)c(spaces)24 b(in)d(whic)m(h)h(the)h (old)e(idea)g(of)g(sequen)m(tial)568 3088 y(limit)29 b(is)j(go)s(o)s(d)f(enough.)324 3421 y Fn(3.1)160 b(The)54 b(F)-13 b(ailure)55 b(of)f(Sequences)324 3640 y Fq(The)31 b(follo)m(wing)c(imp)s(ortan)m(t)i(results)h(are)g(probably)g(familiar) c(to)k(us)h(in)e(the)i(con)m(text)g(of)324 3761 y(metric)g(spaces,)k (or)d(at)g(least)g(in)g(the)h(setting)f(of)g(the)h(real)f(line,)f Fo(R)q Fq(.)324 3989 y Fk(Theorem)37 b(3.1)49 b Fp(Given)36 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))p Fp(,)37 b Fo(A)31 b Fm(\022)h Fo(X)8 b Fp(,)36 b Fo(p)31 b Fm(2)h Fo(X)8 b Fp(:)48 b(if)37 b(ther)-5 b(e)36 b(exists)h(some)e(se)-5 b(quenc)g(e)324 4109 y(of)34 b(p)-5 b(oints)35 b(of)f Fo(A)h Fp(tending)g(to)g Fo(p)p Fp(,)f(then)h Fo(p)28 b Fm(2)1937 4084 y Fq(\026)1912 4109 y Fo(A)p Fp(.)324 4338 y Fk(Theorem)37 b(3.2)49 b Fp(Given)42 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))p Fp(,)45 b Fo(A)d Fm(\022)h Fo(X)8 b Fp(:)60 b(if)43 b Fo(A)g Fp(is)f(close)-5 b(d,)44 b(then)f Fo(A)g Fp(includes)f(the)324 4458 y(limit)34 b(of)h(every)g(c)-5 b(onver)g(gent)34 b(se)-5 b(quenc)g(e)34 b(of)g(p)-5 b(oints)35 b(of)f Fo(A)p Fp(.)324 4686 y Fk(Theorem)j(3.3)49 b Fp(Given)36 b Fo(f)41 b Fq(:)31 b(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))k Fm(!)g Fq(\()p Fo(Y)5 b(;)17 b Fm(T)2090 4627 y Fd(0)2117 4686 y Fq(\))p Fp(:)48 b(if)36 b Fo(f)47 b Fp(is)36 b(c)-5 b(ontinuous,)37 b(then)f Fo(f)47 b Fp(`pr)-5 b(e-)324 4807 y(serves)34 b(limits)h(of)f(se)-5 b(quenc)g(es')34 b(i.e.)45 b(whenever)33 b Fo(x)2125 4822 y Fj(n)2200 4807 y Fm(!)28 b Fo(l)37 b Fp(in)e Fo(X)8 b Fp(,)34 b(then)h Fo(f)11 b Fq(\()p Fo(x)3036 4822 y Fj(n)3083 4807 y Fq(\))28 b Fm(!)f Fo(f)11 b Fq(\()p Fo(l)r Fq(\))35 b Fp(in)324 4927 y Fo(Y)21 b Fp(.)1894 5251 y Fq(27)p eop %%Page: 28 29 28 28 bop 324 548 a Fq(In)43 b(eac)m(h)h(case)g(ab)s(o)m(v)m(e,)i(it)c (is)g(routine)h(to)f(pro)m(v)m(e)i(the)g(statemen)m(t)f(true)g(in)g(a)f (general)324 668 y(top)s(ological)29 b(space)k(as)g(asserted.)45 b(W)-8 b(e)33 b(illustrate)d(b)m(y)k(pro)m(ving)e(Theorem)h(3.3:)324 789 y(Let)38 b Fo(f)49 b Fq(b)s(e)39 b(con)m(tin)m(uous)g(and)f Fo(x)1484 804 y Fj(n)1569 789 y Fm(!)f Fo(l)k Fq(in)c Fo(X)8 b Fq(.)61 b(W)-8 b(e)38 b(m)m(ust)h(sho)m(w)g(that)f Fo(f)11 b Fq(\()p Fo(x)3109 804 y Fj(n)3156 789 y Fq(\))38 b Fm(!)f Fo(f)11 b Fq(\()p Fo(l)r Fq(\).)324 909 y(Giv)m(en)31 b(a)g(neigh)m(b)s(ourho)s(o)s(d)f Fo(N)42 b Fq(of)31 b Fo(f)11 b Fq(\()p Fo(l)r Fq(\),)31 b(there)h(exists)g(op)s(en)g Fo(G)f Fq(suc)m(h)i(that)e Fo(f)11 b Fq(\()p Fo(l)r Fq(\))27 b Fm(2)i Fo(G)e Fm(\022)324 1029 y Fo(N)10 b Fq(.)56 b(Then)37 b Fo(l)g Fm(2)d Fo(f)978 993 y Fg(\000)p Fh(1)1073 1029 y Fq(\()p Fo(G)p Fq(\))g Fm(\022)h Fo(f)1431 993 y Fg(\000)p Fh(1)1525 1029 y Fq(\()p Fo(N)10 b Fq(\))36 b(i.e.)g Fo(f)1945 993 y Fg(\000)p Fh(1)2040 1029 y Fq(\()p Fo(N)10 b Fq(\))36 b(is)g(a)h(neigh)m(b)s(ourho)s(o)s(d)e(of)h Fo(l)j Fq(and)d(so)324 1150 y Fo(x)379 1165 y Fj(n)454 1150 y Fm(2)28 b Fo(f)607 1114 y Fg(\000)p Fh(1)701 1150 y Fq(\()p Fo(N)10 b Fq(\))p Fm(8)p Fo(n)29 b Fm(\025)f 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b(are)d(of)g(immense)f(usefulness)j(in)d(real)324 2835 y(analysis)34 b(and)g(in)g(metric)g(spaces)i(and)e(elsewhere)i(|)e(and) h(their)f(failure)f(to)h(describ)s(e)324 2956 y(general)e(top)s(ology)f (adequately)i(is)f(a)g(tec)m(hnical)h(handicap.)43 b(What)32 b(to)h(do?)324 3288 y Fn(3.2)160 b(Nets)53 b(-)h(A)g(Kind)g(of)g(`Sup)t (er-Sequence')324 3507 y Fq(Recall)33 b(that)i(a)g(sequence)j(is)d (just)g(a)g(function)f(ha)m(ving)h(the)h(p)s(ositiv)m(e)e(in)m(tegers)i (as)f(do-)324 3628 y(main.)43 b(The)34 b(set)g(of)f(p)s(ositiv)m(e)g (in)m(tegers,)g(of)g(course,)h(p)s(ossesses)i(a)d(particularly)e (simple)324 3748 y(ordering;)d(there)g(is)f(a)g(\014rst)g(mem)m(b)s (er,)h(second)g(mem)m(b)s(er,)g(third)f(mem)m(b)s(er,)g(etc.)43 b(Not)27 b(all)324 3869 y(sets)34 b(are)g(naturally)d(endo)m(w)m(ed)36 b(with)d(so)g(simple)f(an)h(ordering.)45 b(F)-8 b(or)33 b(example,)g(dictio-)324 3989 y(nary)d(\(lexographical\))d(ordering)i (of)g(w)m(ords)i(is)e(more)g(complex)g(\(though)h(still)d(relativ)m(e) 324 4109 y(nice)h(as)g(orderings)g(go\).)42 b(By)29 b(replacing)e(the)h (domain)f(of)h(p)s(ositiv)m(e)f(in)m(tegers)i(with)f(a)g(set)324 4230 y(ha)m(ving)k(a)g(more)g(complicated)f(ordering)h(w)m(e)i(will:) 469 4433 y Fm(\017)49 b Fq(de\014ne)33 b(a)g(net)g(\(in)f(analogy)f (with)h(sequence\),)469 4637 y Fm(\017)49 b Fq(iden)m(tify)32 b(an)g(asso)s(ciated)h(notion)e(of)h(con)m(v)m(ergence,)469 4840 y Fm(\017)49 b Fq(sho)m(w)33 b(that)g(net)g(con)m(v)m(ergence)i (is)d(su\016cien)m(t)i(to)e(c)m(haracterize)h(closure)g(of)f(sets,)1894 5251 y(28)p eop %%Page: 29 30 29 29 bop 469 548 a Fm(\017)49 b Fq(and)33 b(that)g(compactness)i(can)f (b)s(e)g(c)m(haracterized)g(in)f(terms)g(of)g(con)m(v)m(ergence)j(of) 568 668 y(subnets.)324 860 y(Note)j(that)g(these)h(last)e(t)m(w)m(o)i (items)e(generalize)g(the)i(role)e(of)g(sequences)43 b(in)38 b(a)h(metric)324 980 y(space.)324 1267 y Fl(3.2.1)136 b(De\014nition)45 b(of)h(Net)324 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b(suc)m(h)h(a)e(case,)i(w)m(e)f(sometimes)f(sa)m(y)h(that)g(the)g(net)g (\()p Fo(x)2406 3440 y Fj(\013)2456 3425 y Fq(\))2494 3440 y Fj(\013)p Fg(2)p Fj(A)324 3546 y Fp(eventuates)j Fo(N)10 b Fq(.)42 b(Clearly)-8 b(,)28 b(this)f(de\014nition)g(incorp)s (orates)h(the)g(old)f(de\014nition)g(of)g(`limit)324 3666 y(of)37 b(a)g(sequence'.)61 b(The)38 b(limit)c(of)j(the)h(net)f Fo(f)49 b Fq(describ)s(ed)38 b(in)e(\(ii\))g(ab)s(o)m(v)m(e)i(is)f(3.) 58 b(In)37 b(\(iii\),)324 3786 y(the)k(net)h(describ)s(ed)g(con)m(v)m (erges)h(to)e Fo(x)h Fq(no)f(matter)f(ho)m(w)i(the)g(v)-5 b(alues)41 b Fo(x)3002 3801 y Fj(N)3111 3786 y Fq(are)g(c)m(hosen)324 3907 y(.)16 b(.)g(.)g(pro)m(v)m(e!)324 4196 y Fl(3.2.3)136 b(Net)45 b(Con)l(v)l(ergence)i(and)d(Closure)324 4380 y Fq(Our)22 b(claim)d(is)i(that)h(nets)g(`fully)f(describ)s(e')h(the)h (structure)g(of)e(a)h(top)s(ological)c(space.)41 b(Our)324 4501 y(\014rst)30 b(piece)h(of)e(evidence)j(to)e(supp)s(ort)g(this)g (is)f(that)h(with)g(nets,)i(instead)e(of)f(sequences,)324 4621 y(Theorems)k(3.1,)f(3.2)h(and)f(3.3)g(ha)m(v)m(e)i(w)m(ork)-5 b(able)33 b(con)m(v)m(erses:)324 4825 y Fk(Theorem)k(3.4)49 b Fp(Given)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))p Fp(,)34 b Fo(A)28 b Fm(\022)g Fo(X)8 b Fp(,)34 b Fo(p)28 b Fm(2)g Fo(X)8 b Fp(:)44 b Fo(p)28 b Fm(2)2512 4799 y Fq(\026)2486 4825 y Fo(A)35 b Fp(i\013)f(ther)-5 b(e)34 b(exists)g(a)g(net)h(in)324 4945 y Fo(A)g Fp(c)-5 b(onver)g(ging)33 b(to)i Fo(p)p Fp(.)1894 5251 y Fq(30)p eop %%Page: 31 32 31 31 bop 324 548 a Fq(Pro)s(of)p 324 561 235 4 v 36 w(If)38 b(some)f(net)g(of)g(p)s(oin)m(ts)g(of)g Fo(A)g Fq(con)m(v)m(erges)i(to)e Fo(p)p Fq(,)i(then)e(ev)m(ery)j(neigh)m(b)s (ourho)s(o)s(d)324 668 y(of)d Fo(p)h Fq(con)m(tains)f(p)s(oin)m(ts)h (of)f Fo(A)g Fq(\(namely)-8 b(,)38 b(v)-5 b(alues)38 b(of)f(the)h(net\))g(and)g(so)g(w)m(e)h(get)e Fo(p)g Fm(2)3487 643 y Fq(\026)3461 668 y Fo(A)q Fq(.)324 789 y(Con)m(v)m(ersely)-8 b(,)34 b(if)c Fo(p)i 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Fk(De\014nition)h(3.7)49 b Fp(L)-5 b(et)38 b Fq(\()p Fo(x)1303 3001 y Fj(\013)1353 2986 y Fq(\))1391 3001 y Fj(\013)p Fg(2)p Fj(A)1577 2986 y Fp(and)f Fq(\()p Fo(y)1855 3001 y Fj(\014)1902 2986 y Fq(\))1940 3001 y Fj(\014)s Fg(2)p Fj(B)2128 2986 y Fp(b)-5 b(e)37 b(any)h(two)f(nets.) 53 b(We)37 b(c)-5 b(al)5 b(l)37 b Fq(\()p Fo(y)3326 3001 y Fj(\014)3373 2986 y Fq(\))3411 3001 y Fj(\014)s Fg(2)p Fj(B)324 3107 y Fp(a)k Fq(subnet)h Fp(of)e Fq(\()p Fo(x)951 3122 y Fj(\013)1001 3107 y Fq(\))1039 3122 y Fj(\013)p Fg(2)p Fj(A)1229 3107 y Fp(pr)-5 b(ovide)g(d)40 b(that)i(every)e(tail)h (of)g Fq(\()p Fo(x)2481 3122 y Fj(\013)2531 3107 y Fq(\))g Fp(c)-5 b(ontains)40 b(a)h(tail)f(of)h Fq(\()p Fo(y)3477 3122 y Fj(\014)3524 3107 y Fq(\))324 3227 y Fp(i.e.)j(pr)-5 b(ovide)g(d:)921 3441 y Fm(8)p Fo(\013)1038 3456 y Fh(0)1105 3441 y Fm(2)28 b Fo(A)17 b Fm(9)p Fo(\014)1399 3456 y Fh(0)1467 3441 y Fm(2)28 b Fo(B)40 b Fp(such)34 b(that)i Fo(x)p Fq(\()p Fo(\013)2251 3456 y Fh(0)2318 3441 y Fm(!)p Fq(\))27 b Fm(\023)i Fo(y)t Fq(\()p Fo(\014)2734 3456 y Fh(0)2800 3441 y Fm(!)p Fq(\))p Fo(:)324 3662 y Fq(W)-8 b(e)35 b(exp)s(ected)i(a)d(de\014nition)g(lik)m(e)g(`subsequence')k(to) d(turn)f(up)i(here)f(and)g(w)m(e)h(are)e(dis-)324 3783 y(app)s(oin)m(ted)e(that)g(it)g(has)h(to)f(b)s(e)h(so)g(complicated.) 324 3903 y(Net)j(theory)g(ceases)h(to)e(b)s(e)h(a)f(straigh)m(tforw)m (ard)g(generalisation)f(of)h(sequence)j(theory)324 4023 y(precisely)25 b(when)h(w)m(e)g(ha)m(v)m(e)g(to)f(tak)m(e)g(a)g(subnet) h(.)16 b(.)g(.)g(so)26 b(w)m(e'll)e(try)h(to)g(a)m(v)m(oid)g(this)f (whenev)m(er)324 4144 y(p)s(ossible!)43 b(There)33 b(is)g(ho)m(w)m(ev)m (er)h(one)f(result)g(certainly)f(w)m(orth)h(kno)m(wing:)324 4342 y Fk(Theorem)k(3.7)49 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))35 b Fp(is)f(c)-5 b(omp)g(act)33 b(i\013)h(in)g Fo(X)8 b Fp(,)35 b(every)f(net)g(has)g(\(at)h(le)-5 b(ast)34 b(one\))f(c)-5 b(on-)324 4462 y(ver)g(gent)34 b(subnet.)324 4660 y Fq(\(So,)e(for)g(example,)h(\()p Fo(n)p 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Fj(\014)914 2939 y Fq(\))f(b)s(e)g(a)g(subnet)i(of)e(\()p Fo(x)1756 2954 y Fj(\013)1805 2939 y Fq(\);)45 b(let)39 b Fo(N)51 b Fq(b)s(e)40 b(a)g(neigh)m(b)s(ourho)s(o)s(d)g(of)g Fo(l)r Fq(.)66 b(Then)324 3059 y(there)28 b(exists)h Fo(\013)895 3074 y Fh(0)962 3059 y Fq(suc)m(h)g(that)e Fo(x)1438 3074 y Fj(\013)1516 3059 y Fm(2)h Fo(N)38 b Fq(for)27 b(all)f Fo(\013)i Fm(\025)g Fo(\013)2258 3074 y Fh(0)2298 3059 y Fq(.)41 b(F)-8 b(urther,)29 b(there)g(exists)f Fo(\014)3307 3074 y Fh(0)3374 3059 y Fq(suc)m(h)324 3180 y(that)k Fm(f)p Fo(y)633 3195 y Fj(\014)707 3180 y Fq(:)c Fo(\014)33 b Fm(\025)c Fo(\014)1011 3195 y Fh(0)1050 3180 y Fm(g)f(\022)g(f)p Fo(x)1338 3195 y Fj(\013)1415 3180 y Fq(:)g Fo(\013)g Fm(\025)g Fo(\013)1727 3195 y Fh(0)1767 3180 y Fm(g)k Fq(and)h(so)g Fo(y)2207 3195 y Fj(\014)2281 3180 y Fm(2)28 b Fo(N)43 b Fq(for)32 b(all)f Fo(\014)i Fm(\025)28 b Fo(\014)3029 3195 y Fh(0)3069 3180 y Fq(.)324 3506 y Fn(3.3)160 b(First)37 b(Coun)l(table)d(Spaces)h (-)h(Where)e(Sequences)691 3688 y(Su\016ce)324 3907 y Fq(Wh)m(y)45 b(do)f(sequences)k(su\016ce)e(to)e(describ)s(e)g (structure)i(in)d Fo(R)q Fq(,)48 b Fo(C)j Fq(and)44 b(other)h(metric) 324 4028 y(spaces)28 b(but)g(not)f(in)f(man)m(y)h(other)g(top)s (ological)d(spaces?)43 b(The)28 b(k)m(ey)g(here)g(is)f(recognizing)324 4148 y(that)d(man)m(y)g(pro)s(ofs)g(regarding)f(con)m(v)m(ergence)k(in) c(metric)g(spaces)j(in)m(v)m(olv)m(e)e(constructing)324 4269 y(sequences)39 b(of)d(nested)i(op)s(en)f(sets)g(ab)s(out)f(a)g(p)s (oin)m(t.)54 b(Sometimes)35 b(these)j(describ)s(e)f(the)324 4389 y(top)s(ological)26 b(structure)31 b(near)g(the)f(p)s(oin)m(t)f (and)h(other)h(times)e(not.)42 b(In)30 b(what)h(follo)m(ws)e(w)m(e)469 4552 y Fm(\017)49 b Fq(iden)m(tify)37 b(the)i(lo)s(cal)d(c)m (haracteristic)i(of)g(top)s(ological)d(space)k(that)f(mak)m(es)h(these) 568 4672 y(pro)s(ofs)32 b(w)m(ork,)469 4862 y Fm(\017)49 b Fq(and)38 b(pro)m(v)m(e)h(that)f(sequences)j(su\016ce)e(to)f(describ) s(e)h(the)f(top)s(ological)c(structure)568 4983 y(of)e(spaces)i(with)e (this)g(c)m(haracteristic.)1894 5251 y(32)p eop %%Page: 33 34 33 33 bop 324 548 a Fl(3.3.1)136 b(First)45 b(Coun)l(table)h(Spaces)324 733 y Fq(So)36 b(what)h(c)m(haracteristic)g(common)e(to)i Fo(R)q Fq(,)h Fo(C)43 b Fq(and)37 b(other)g(metric)f(spaces)i(mak)m(es) f(se-)324 853 y(quences)e(so)d(`go)s(o)s(d')g(at)g(describing)h(their)f (structure?)324 1021 y Fk(De\014nition)k(3.8)49 b Fp(L)-5 b(et)42 b Fo(x)f Fm(2)h Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))p Fp(.)66 b(A)43 b Fk(coun)m(table)j(neigh)m(b)s(ourho)s(o)s(d) h(base)h(at)324 1142 y Fo(x)37 b Fp(me)-5 b(ans:)47 b(a)36 b(se)-5 b(quenc)g(e)36 b Fo(N)1320 1157 y Fh(1)1360 1142 y Fp(,)g Fo(N)1504 1157 y Fh(2)1544 1142 y Fp(,)h Fo(N)1689 1157 y Fh(3)1728 1142 y Fp(,)g(.)15 b(.)g(.)g(of)36 b(p)-5 b(articular)37 b(neighb)-5 b(ourho)g(o)g(ds)35 b(of)h Fo(x)h Fp(such)324 1262 y(that)e(every)g(neighb)-5 b(ourho)g(o)g(d)33 b(of)i Fo(x)g Fp(shal)5 b(l)34 b(c)-5 b(ontain)34 b(one)h(of)f(the)h Fo(N)2731 1277 y Fj(i)2760 1262 y Fp('s.)324 1430 y Fq(Note)30 b(that)g(w)m(e)i(ma)m(y)e(assume)h(that)f Fo(N)1739 1445 y Fh(1)1806 1430 y Fm(\023)e Fo(N)1989 1445 y Fh(2)2056 1430 y Fm(\023)h Fo(N)2240 1445 y Fh(3)2307 1430 y Fm(\023)f(\001)17 b(\001)g(\001)28 b Fq(b)s(ecause,)33 b(if)c(not,)i(then)g(w)m(e)324 1551 y(can)i(w)m(ork)g(with)f Fo(N)1041 1566 y Fh(1)1081 1551 y Fq(,)g Fo(N)1218 1566 y Fh(1)1280 1551 y Fm(\\)23 b Fo(N)1447 1566 y Fh(2)1486 1551 y Fq(,)33 b Fo(N)1624 1566 y Fh(1)1686 1551 y Fm(\\)22 b Fo(N)1852 1566 y Fh(2)1914 1551 y Fm(\\)g Fo(N)2080 1566 y Fh(3)2120 1551 y Fq(,)33 b(.)16 b(.)g(.)324 1732 y Fk(De\014nition)36 b(3.9)49 b Fp(We)32 b(c)-5 b(al)5 b(l)32 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))32 b Fk(\014rst-coun)m(table)g Fp(when)g(every)g(p)-5 b(oint)32 b(in)f Fo(X)40 b Fp(has)324 1853 y(a)34 b(c)-5 b(ountable)35 b(neighb)-5 b(ourho)g(o)g(d)33 b(b)-5 b(ase.)324 2034 y(Example)52 b Fq(The)c(classic)d(example)h(of)g(a)g(\014rst-coun) m(table)g(space)h(is)f(an)m(y)h(metric)e(\(or)324 2154 y(metrizable\))29 b(space)j(b)s(ecause)h(if)c Fo(x)g Fm(2)f Fq(\()p Fo(M)5 b(;)17 b(d)p Fq(\),)31 b(then)g Fo(B)5 b Fq(\()p Fo(x;)17 b Fq(1\),)32 b Fo(B)5 b Fq(\()p Fo(x;)2857 2115 y Fh(1)p 2857 2131 36 4 v 2857 2189 a(2)2902 2154 y Fq(\),)31 b Fo(B)5 b Fq(\()p Fo(x;)3224 2115 y Fh(1)p 3224 2131 V 3224 2189 a(3)3270 2154 y Fq(\),)31 b(.)16 b(.)g(.)g(is)324 2275 y(a)32 b(coun)m(table)h(neigh)m(b)s(ourho) s(o)s(d)e(base)j(at)e Fo(x)p Fq(.)324 2443 y Fk(Theorem)37 b(3.8)49 b Fp(First-c)-5 b(ountability)43 b(is)f(her)-5 b(e)g(ditary)43 b(and)f(pr)-5 b(eserve)g(d)42 b(by)h(c)-5 b(ontinuous)324 2563 y(op)g(en)34 b(onto)h(maps.)324 2732 y Fq(Pro)s(of)p 324 2745 235 4 v 32 w(Left)d(to)g(the)h(reader.) 324 2900 y Fk(Theorem)k(3.9)134 b Fp(\(i\))49 b(Complete)34 b(sep)-5 b(ar)g(ability)34 b(implies)g(\014rst)h(c)-5 b(ountability.)379 3092 y(\(ii\))49 b(Converse)33 b(not)i(always)g (true.)350 3283 y(\(iii\))48 b(Converse)33 b(valid)h(on)h(a)g(c)-5 b(ountable)34 b(underlying)g(set.)324 3452 y Fq(Pro)s(of)p 324 3465 V 416 3620 a(\(i\))48 b(If)23 b Fm(B)k Fq(is)c(a)h(coun)m (table)f(base)i(for)e(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))f(and)f Fo(p)28 b Fm(2)g Fo(X)8 b Fq(,)25 b(consider)f Fm(f)p Fo(B)33 b Fm(2)28 b(B)j Fq(:)d Fo(p)g Fm(2)g Fo(B)5 b Fm(g)568 3740 y Fq(whic)m(h)41 b(is)g(a)g(coun)m(table)h(family)d(of)h (neigh)m(b)s(ourho)s(o)s(ds)h(of)g Fo(p)p Fq(.)70 b(Moreo)m(v)m(er,)45 b(they)568 3861 y(form)31 b(a)h(neigh)m(b)s(ourho)s(o)s(d)g(base)h(at)f Fo(p)p Fq(.)389 4052 y(\(ii\))47 b(An)27 b(uncoun)m(table)g(discrete)g (space)h(is)e(\014rst)i(coun)m(table)f(\()f(since)h(metrizable\),)g(y)m (et)568 4173 y(is)32 b(not)g(completely)g(separable.)362 4365 y(\(iii\))46 b(Supp)s(ose)f Fo(X)53 b Fq(coun)m(table)45 b(and)g(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))45 b(\014rst)g(coun)m (table.)80 b(F)-8 b(or)44 b(eac)m(h)i Fo(x)j Fm(2)g Fo(X)8 b Fq(,)568 4485 y(c)m(ho)s(ose)28 b(a)g(coun)m(table)g(neigh)m(b)s (ourho)s(o)s(d)e(base:)42 b Fo(N)10 b Fq(\()p Fo(x;)17 b Fq(1\),)29 b Fo(N)10 b Fq(\()p Fo(x;)17 b Fq(2\),)29 b Fo(N)10 b Fq(\()p Fo(x;)17 b Fq(3\),)29 b(.)16 b(.)g(.)g(.)568 4605 y(Eac)m(h)34 b(is)f(a)g(neigh)m(b)s(ourho)s(o)s(d)f(of)h Fo(x)h Fq(and)g(so)f(con)m(tains)h(an)f(op)s(en)h(neigh)m(b)s(ourho)s (o)s(d)568 4726 y(of)e Fo(x)p Fq(:)44 b Fo(G)p Fq(\()p Fo(x;)17 b Fq(1\),)32 b Fo(G)p Fq(\()p Fo(x;)17 b Fq(2\),)32 b Fo(G)p Fq(\()p Fo(x;)17 b Fq(3\),)33 b(.)16 b(.)g(.)g(.)568 4882 y(Then)33 b Fm(B)f Fq(=)27 b Fm(f)p Fo(G)p Fq(\()p Fo(x;)17 b(n)p Fq(\))28 b(:)g Fo(n)g Fm(2)g Fo(N)5 b(;)34 b(x)28 b Fm(2)g Fo(X)8 b Fm(g)32 b Fq(is)g(a)h(coun)m(table)f(family)e (of)j(op)s(en)f(sets)568 5002 y(and)g(is)g(a)h(base)g(for)f(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\).)43 b(Th)m(us,)34 b(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))32 b(is)g(completely)g(separable.)1894 5251 y(33)p eop %%Page: 34 35 34 34 bop 324 548 a Fp(Example)324 668 y Fq(The)25 b(Arens-F)-8 b(ort)24 b(space)i(\(see,)h(for)d(example,)h(Steen)h(and)e(Seebac)m(h,) k Fp(Counter)-5 b(examples)324 789 y(in)37 b(T)-7 b(op)i(olo)g(gy)42 b Fq(is)35 b(not)g(\014rst-coun)m(table)h(b)s(ecause)g(otherwise)g(it)e (w)m(ould)h(b)s(e)h(completely)324 909 y(separable)c(whic)m(h)i(is)e (false!)324 1198 y Fl(3.3.2)136 b(P)l(o)l(w)l(er)46 b(of)f(Sequences)g (in)g(First)g(Coun)l(table)h(Spaces)324 1383 y Fq(The)24 b(follo)m(wing)d(three)k(results)f(illustrate)d(that)j(`sequences)j (su\016ce)e(for)e(\014rst-coun)m(table)324 1503 y(spaces')k(in)e(the)i (sense)g(that)f(w)m(e)h(don't)f(need)h(to)e(use)i(nets)g(to)e(describ)s (e)i(their)e(structure.)324 1623 y(This)37 b(is)g(wh)m(y)i(sequences)i (are)c(su\016cien)m(tly)h(general)f(to)g(describ)s(e,)j(fully)-8 b(,)37 b(metric)f(and)324 1744 y(metrizable)31 b(spaces.)324 1972 y Fk(Theorem)37 b(3.10)49 b Fp(Given)34 b(a)h(\014rst-c)-5 b(ountable)35 b(sp)-5 b(ac)g(e)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))409 2176 y Fp(\(i\))49 b Fo(p)36 b Fm(2)h Fo(X)8 b Fp(,)40 b Fo(A)d Fm(\022)g Fo(X)8 b Fp(,)40 b(then)g Fo(p)c Fm(2)1734 2150 y Fq(\026)1708 2176 y Fo(A)k Fp(i\013)f(ther)-5 b(e)39 b(exists)h(a)f(se)-5 b(quenc)g(e)p 2558 2208 362 4 v 39 w(of)39 b(p)-5 b(oints)39 b(of)h Fo(A)568 2296 y Fp(c)-5 b(onver)g(ging)33 b(to)i Fo(p)p Fp(.)379 2499 y(\(ii\))49 b Fo(A)27 b Fm(\022)i Fo(X)38 b Fp(is)30 b(close)-5 b(d)30 b(i\013)g Fo(A)h Fp(c)-5 b(ontains)29 b(every)i(limit)f(of)g(every)h(c)-5 b(onver)g(gent)29 b(se)-5 b(quenc)g(e)p 3200 2532 V 568 2620 a(of)34 b(its)h(own)f(p)-5 b(oints.)350 2823 y(\(iii\))48 b Fo(f)38 b Fq(:)28 b(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))j Fm(!)f Fq(\()p Fo(Y)5 b(;)17 b Fm(T)1370 2764 y Fd(0)1396 2823 y Fq(\))28 b Fp(is)f(c)-5 b(ontinuous)27 b(i\013)h(it)g(pr)-5 b(eserves)26 b(limits)i(of)f(\(c)-5 b(onver)g(gent\))568 2943 y(se)g(quenc)g(es)p 568 2976 402 4 v -1 w(.)324 3172 y Fq(Pro)s(of)p 324 3185 235 4 v 416 3375 a(\(i\))48 b(Theorem)43 b(3.1)g(said)g(that)g(if)f(there)i(exists)g(a)f(sequence)j (in)c Fo(A)h Fq(con)m(v)m(erging)h(to)568 3496 y(some)32 b Fo(p)c Fm(2)g Fo(X)8 b Fq(,)32 b(then)h Fo(p)28 b Fm(2)1550 3470 y Fq(\026)1524 3496 y Fo(A)p Fq(.)568 3657 y(Con)m(v)m(ersely)-8 b(,)42 b(if)c Fo(p)g Fm(2)1402 3632 y Fq(\026)1376 3657 y Fo(A)p Fq(,)i(then)g Fo(p)f Fq(has)g(a)f(coun)m(table)h(base)h(of)e (neigh)m(b)s(ourho)s(o)s(ds)568 3778 y Fo(N)646 3793 y Fh(1)735 3778 y Fm(\023)50 b Fo(N)940 3793 y Fh(2)1030 3778 y Fm(\023)g Fo(N)1235 3793 y Fh(3)1324 3778 y Fm(\023)h(\001)17 b(\001)g(\001)n Fq(,)49 b(eac)m(h)d(of)f(whic)m(h)h(m)m(ust)g(in)m (tersect)h Fo(A)p Fq(.)82 b(So)46 b(c)m(ho)s(ose)568 3898 y Fo(x)623 3913 y Fj(j)687 3898 y Fm(2)28 b Fo(N)859 3913 y Fj(j)916 3898 y Fm(\\)21 b Fo(A)32 b Fq(for)f(all)e Fo(j)34 b Fm(\025)28 b Fq(1.)43 b(Then)33 b(\()p Fo(x)2035 3913 y Fj(j)2072 3898 y Fq(\))e(is)h(a)f(sequence)j(in)d Fo(A)h Fq(and,)g(giv)m(en)g(an)m(y)568 4019 y(neigh)m(b)s(ourho)s(o)s (d)e Fo(H)38 b Fq(of)30 b Fo(p)p Fq(,)i Fo(H)38 b Fq(m)m(ust)31 b(con)m(tain)f(one)h(of)g(the)g Fo(N)2801 4034 y Fj(j)2838 4019 y Fq('s)g(i.e.)g Fo(H)k Fm(\023)28 b Fo(N)3389 4034 y Fj(j)3418 4043 y Fe(0)3484 4019 y Fm(\023)568 4139 y Fo(N)646 4154 y Fj(j)675 4163 y Fe(0)709 4154 y Fh(+1)831 4139 y Fm(\023)g(\001)17 b(\001)g(\001)31 b Fq(so)i(that)f Fo(x)1471 4154 y Fj(j)1536 4139 y Fm(2)c Fo(H)40 b Fq(for)32 b(all)e Fo(j)k Fm(\025)28 b Fo(j)2254 4154 y Fh(0)2294 4139 y Fq(.)43 b(That)33 b(is,)f Fo(x)2788 4154 y Fj(j)2853 4139 y Fm(!)27 b Fo(p)p Fq(.)389 4342 y(\(ii\))47 b(Corollary)31 b(of)h(\(i\).)362 4546 y(\(iii\))46 b Fo(f)39 b Fq(con)m(tin)m(uous)29 b(implies)d(that)i(it)f(m)m(ust)i(preserv)m(e)h(limits)c(of)h (sequences)32 b(\(b)m(y)d(The-)568 4666 y(orem)j(3.3\).)43 b(Con)m(v)m(ersely)-8 b(,)35 b(if)d Fo(f)43 b Fq(is)33 b(not)f(con)m(tin)m(uous,)i(there)f(exists)h Fo(A)28 b Fm(\022)h Fo(X)40 b Fq(suc)m(h)568 4787 y(that)f Fo(f)11 b Fq(\()908 4761 y(\026)883 4787 y Fo(A)p Fq(\))39 b Fm(6\022)p 1149 4702 208 4 v 39 w Fo(f)11 b Fq(\()p Fo(A)p Fq(\).)64 b(Th)m(us,)43 b(there)d(exists)g Fo(p)f Fm(2)h Fo(f)11 b Fq(\()2579 4761 y(\026)2554 4787 y Fo(A)o Fq(\))27 b Fm(n)p 2768 4702 V 27 w Fo(f)11 b Fq(\()p Fo(A)p Fq(\))39 b(so)g Fo(p)g Fq(=)h Fo(f)11 b Fq(\()p Fo(x)p Fq(\),)568 4907 y(some)32 b Fo(x)c Fm(2)1015 4882 y Fq(\026)989 4907 y Fo(A)p Fq(.)44 b(So)32 b(there)i(exists)f(a)f(sequence)k(\()p Fo(x)2366 4922 y Fj(n)2413 4907 y Fq(\))c(in)g Fo(A)h Fq(with)f Fo(x)2980 4922 y Fj(n)3055 4907 y Fm(!)27 b Fo(x)p Fq(.)1894 5251 y(34)p eop %%Page: 35 36 35 35 bop 568 548 a Fq(Y)-8 b(et,)33 b(if)f Fo(f)11 b Fq(\()p Fo(x)1016 563 y Fj(n)1063 548 y Fq(\))29 b Fm(!)f Fo(f)11 b Fq(\()p Fo(x)p Fq(\)\(=)29 b Fo(p)p Fq(\),)k Fo(p)g Fq(w)m(ould)g(b)s(e)h(the)f(limit)d(of)j(a)g(sequence)j(in)c Fo(f)11 b Fq(\()p Fo(A)p Fq(\))568 668 y(so)29 b(that)h Fo(p)d Fm(2)p 1064 584 208 4 v 29 w Fo(f)11 b Fq(\()p Fo(A)p Fq(\))29 b(|con)m(tradiction!)41 b(Th)m(us)31 b Fo(f)40 b Fq(fails)28 b(to)h(preserv)m(e)j(con)m(v)m(ergence)568 789 y(of)g(this)g(sequence.)1894 5251 y(35)p eop %%Page: 36 37 36 36 bop 324 1212 a Fr(Chapter)78 b(4)324 1627 y(Pro)6 b(duct)78 b(Spaces)324 2080 y Fq(A)26 b(common)f(task)i(in)e(top)s (ology)g(is)h(to)g(construct)h(new)g(top)s(ological)c(spaces)k(from)e (other)324 2200 y(spaces.)59 b(One)38 b(w)m(a)m(y)g(of)f(doing)g(this)g (is)g(b)m(y)h(taking)e(pro)s(ducts.)59 b(All)35 b(are)j(familiar)33 b(with)324 2320 y(iden)m(tifying)41 b(the)j(plane)e(or)h(3-dimensional) d(Euclidean)j(space)h(with)e(ordered)i(pairs)324 2441 y(or)35 b(triples)g(of)g(n)m(um)m(b)s(ers)h(eac)m(h)h(of)e(whic)m(h)h (is)f(a)g(mem)m(b)s(er)g(of)g(the)h(real)f(line.)51 b(F)-8 b(ew)m(er)37 b(are)324 2561 y(probably)23 b(familar)d(with)j(realizing) e(the)j(torus)g(as)g(ordered)g(pairs)f(of)g(complex)g(n)m(um)m(b)s(ers) 324 2682 y(of)32 b(mo)s(dulus)f(one.)44 b(In)33 b(this)f(c)m(hap)s(er)i (w)m(e)f(answ)m(er)h(t)m(w)m(o)f(questions:)469 2885 y Fm(\017)49 b Fq(Ho)m(w)22 b(do)g(the)g(ab)s(o)m(v)m(e)g(pro)s(duct)g (constructions)h(generalize)e(to)g(top)s(ological)d(spaces?)469 3088 y Fm(\017)49 b Fq(What)32 b(top)s(ological)d(prop)s(erties)k(are)f (preserv)m(ed)j(b)m(y)f(this)e(construction?)324 3421 y Fn(4.1)160 b(Constructing)52 b(Pro)t(ducts)324 3640 y Fq(The)33 b(pro)s(cess)h(of)e(constructing)h(a)f(pro)s(duct)h(falls)e (naturally)g(in)m(to)h(t)m(w)m(o)h(stages.)469 3844 y Fm(\017)49 b Fq(The)29 b(\014rst)h(stage,)f(whic)m(h)h(is)e(en)m (tirely)g(set-theoretic,)i(consists)f(in)f(describing)h(an)568 3964 y(elemen)m(t)41 b(of)f(the)i(underlying)e(set)i(of)f(the)h(pro)s (duct.)69 b(This)41 b(task)h(is)f(primarily)568 4084 y(one)33 b(of)f(generalizing)e(the)j(notion)f(of)g(ordered)h(pair)e(or) i(triple.)469 4288 y Fm(\017)49 b Fq(The)38 b(second)g(stage)g(is)e (describing)h(what)g(op)s(en)h(sets)g(lo)s(ok)e(lik)m(e.)56 b(This)38 b(will)c(b)s(e)568 4408 y(done)f(b)m(y)i(describing)e(a)g (subbasis)h(for)f(the)g(top)s(ology)-8 b(.)45 b(The)34 b(guiding)d(goal)h(is)h(to)568 4529 y(pro)m(vide)38 b(just)g(enough)h (op)s(ens)f(sets)i(to)d(guaran)m(tee)i(the)f(con)m(tin)m(uit)m(y)g(of)g (certain)568 4649 y(imp)s(ortan)m(t)30 b(functions.)1894 5251 y(36)p eop %%Page: 37 38 37 37 bop 324 548 a Fl(4.1.1)136 b(Set-Theoretic)45 b(Construction)324 733 y Fq(Supp)s(ose)23 b(throughout)e(that)h(w)m(e)g(are)g(giv)m(en)g (a)g(family)d(of)i(top)s(ological)d(spaces)23 b Fm(f)p Fq(\()p Fo(X)3323 748 y Fj(i)3351 733 y Fo(;)17 b Fm(T)3449 748 y Fj(i)3477 733 y Fq(\))28 b(:)324 853 y Fo(i)g Fm(2)g Fo(I)8 b Fm(g)32 b Fq(where)i Fo(I)40 b Fq(is)32 b(some)h(non-empt)m(y) f(`lab)s(elling')e(or)i(index)h(set.)324 973 y(Our)25 b(\014rst)h(task)g(is)f(to)g(get)g(a)g(clear)g(men)m(tal)f(picture)i (of)e(what)i(w)m(e)g(mean)f(b)m(y)i(the)e(pro)s(duct)324 1094 y(of)35 b(the)i Fp(sets)e Fo(X)883 1109 y Fj(i)911 1094 y Fq(.)54 b(Lo)s(ok)35 b(again)g(at)g(the)i(\014nite)e(case)i (where)g Fo(I)42 b Fq(=)33 b Fm(f)p Fq(1)p Fo(;)17 b Fq(2)p Fo(;)g(:)g(:)g(:)e(;)i(n)p Fm(g)p Fq(.)53 b(Here,)324 1214 y(the)33 b(pro)s(duct)g(set)339 1486 y Fo(X)j Fq(=)27 b Fo(X)640 1501 y Fh(1)702 1486 y Fm(\002)22 b Fo(X)882 1501 y Fh(2)944 1486 y Fm(\002)g Fo(X)1124 1501 y Fh(3)1186 1486 y Fm(\002)h Fo(:)17 b(:)g(:)k Fm(\002)i Fo(X)1603 1501 y Fj(n)1677 1486 y Fq(=)1817 1378 y Fj(n)1785 1403 y Ff(Y)1781 1586 y Fj(i)p Fh(=1)1912 1486 y Fo(X)1993 1501 y Fj(i)2049 1486 y Fq(=)k Fm(f)p Fq(\()p Fo(p)2289 1501 y Fh(1)2328 1486 y Fo(;)17 b(p)2421 1501 y Fh(2)2460 1486 y Fo(;)g(:)g(:)g(:)f(;)h(p)2728 1501 y Fj(n)2775 1486 y Fq(\))27 b(:)h Fo(p)2944 1501 y Fj(i)3000 1486 y Fm(2)g Fo(X)3175 1501 y Fj(i)3203 1486 y Fo(;)34 b(i)27 b Fm(2)i Fo(I)8 b Fm(g)p Fo(:)324 1777 y Fq(i.e.)42 b(the)30 b(elemen)m(ts)g(of)f Fo(X)37 b Fq(are)29 b(the)h(functions)g Fo(x)e Fq(:)g Fo(I)35 b Fm(!)28 b([)2430 1740 y Fj(n)2430 1801 y(i)p Fh(=1)2548 1777 y Fo(X)2629 1792 y Fj(i)2687 1777 y Fq(suc)m(h)j(that)e Fo(x)p Fq(\(1\))f Fm(2)g Fo(X)3495 1792 y Fh(1)3535 1777 y Fq(,)324 1897 y Fo(x)p Fq(\(2\))g Fm(2)g Fo(X)707 1912 y Fh(2)746 1897 y 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b(that)e(w)m(e)i(normally)c(write)i Fo(x)1698 2568 y Fj(i)1760 2553 y Fq(rather)g(than)h Fo(x)p Fq(\()p Fo(i)p Fq(\).)324 2673 y(Then)k(a)e(t)m(ypical)g(elemen)m(t)h(of)f Fo(X)41 b Fq(=)1697 2607 y Ff(Q)1792 2673 y Fo(X)1873 2688 y Fj(i)1937 2673 y Fq(will)34 b(lo)s(ok)g(lik)m(e:)49 b(\()p Fo(x)2655 2688 y Fj(i)2684 2673 y Fq(\))2722 2688 y Fj(i)p Fg(2)p Fj(I)2868 2673 y Fq(or)36 b(just)g(\()p Fo(x)3280 2688 y Fj(i)3308 2673 y Fq(\).)53 b(W)-8 b(e)324 2812 y(will)24 b(still)g(call)g Fo(x)913 2827 y Fj(i)968 2812 y Fq(the)i Fo(i)1162 2776 y Fq(th)1285 2812 y(co)s(ordinate)f(of)h (\()p Fo(x)1956 2827 y Fj(i)1985 2812 y Fq(\))2023 2827 y Fj(i)p Fg(2)p Fj(I)2133 2812 y Fq(.)42 b(\()26 b(Note)g(that)g(the)h (Axiom)e(of)g(Choice)324 2932 y(assures)34 b(us)f(that)993 2866 y Ff(Q)1088 2932 y Fo(X)1169 2947 y Fj(i)1229 2932 y Fq(is)f(non-empt)m(y)h(pro)m(vided)g(none)g(of)f(the)h Fo(X)2809 2947 y Fj(i)2837 2932 y Fq('s)g(are)g(empt)m(y)-8 b(.\))324 3221 y Fl(4.1.2)136 b(T)-11 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Fo(\031)3227 4627 y Fg(\000)p Fh(1)3223 4691 y Fj(i)3247 4701 y Fi(j)3321 4668 y Fq(\()p Fo(G)3436 4683 y Fj(i)3460 4693 y Fi(j)3497 4668 y Fq(\).)324 4788 y(The)k(only)f(dra)m(w)m(able)h(case)h Fo(I)h Fq(=)28 b Fm(f)p Fq(1)p Fo(;)17 b Fq(2)p Fm(g)31 b Fq(ma)m(y)i(help)f(explain:)324 4909 y(\(Here)h(will)d(b)s(e,)j(ev)m (en)m(tually)-8 b(,)33 b(a)f(picture!\))1894 5251 y(37)p eop %%Page: 38 39 38 38 bop 324 548 a Fq(W)-8 b(e)47 b(use)h(these)g(op)s(en)g(cylinders) f(and)g(b)s(o)m(xes)h(to)f(generate)g(a)g(top)s(ology)f(with)g(just)324 668 y(enough)51 b(op)s(en)f(sets)i(to)e(guaran)m(tee)h(that)f(pro)5 b(jection)51 b(maps)f(will)e(b)s(e)j(con)m(tin)m(uous.)324 789 y(Note)42 b(that)h(the)g(op)s(en)g(cylinders)f(form)g(a)g(subbase)i (for)e(a)g(certain)h(top)s(ology)e Fm(T)68 b Fq(on)324 909 y Fo(X)42 b Fq(=)557 843 y Ff(Q)652 909 y Fo(X)733 924 y Fj(i)798 909 y Fq(and)36 b(therefore)i(the)e(op)s(en)h(b)s(o)m (xes)h(form)d(a)h(base)i(for)e Fm(T)25 b Fq(;)39 b Fm(T)62 b Fq(is)36 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Fm(T)1579 2509 y Fj(n)1656 2494 y Fq(is)h(the)h(usual)f(top)s(ology)f (on)i Fo(R)2774 2458 y Fj(n)2821 2494 y Fq(,)g(and)g Fm(T)55 b Fq(the)30 b(usual)324 2614 y(top)s(ology)h(on)h Fo(R)q Fq(,)h(then)1107 2830 y(\()p Fo(R)q(;)17 b Fm(T)25 b Fq(\))d Fm(\002)h Fq(\()p Fo(R)q(;)17 b Fm(T)25 b Fq(\))d Fm(\002)h Fo(:)17 b(:)g(:)f Fq(\()p Fo(R)q(;)h Fm(T)25 b Fq(\))j(=)g(\()p Fo(R)2549 2789 y Fj(n)2596 2830 y Fo(;)17 b Fm(T)2694 2845 y Fj(n)2740 2830 y Fq(\))324 3046 y(as)33 b(one)f(w)m(ould)h(hop)s(e!)324 3245 y Fk(Lemma)k(4.1)49 b Fp(In)37 b(a)h(pr)-5 b(o)g(duct)38 b(sp)-5 b(ac)g(e)38 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))p Fp(,)39 b Fo(N)48 b Fp(is)38 b(a)g(neighb)-5 b(ourho)g(o)g(d)37 b(of)h Fo(p)33 b Fm(2)h Fo(X)46 b Fp(i\013)324 3366 y(ther)-5 b(e)35 b(exists)f(some)g(op)-5 b(en)34 b(b)-5 b(ox)35 b Fo(B)40 b Fp(such)34 b(that)i Fo(p)27 b Fm(2)h Fo(B)33 b Fm(\022)28 b Fo(N)10 b Fp(.)324 3589 y Fk(Lemma)37 b(4.2)49 b Fp(F)-7 b(or)34 b(e)-5 b(ach)34 b Fo(i)28 b Fm(2)g Fo(I)8 b Fp(,)409 3789 y(\(i\))49 b Fo(\031)623 3804 y Fj(i)686 3789 y Fp(is)35 b(c)-5 b(ontinuous)379 3991 y(\(ii\))49 b Fo(\031)623 4006 y Fj(i)686 3991 y Fp(is)35 b(an)f(op)-5 b(en)34 b(mapping.)324 4214 y Fq(Pro)s(of)p 324 4227 235 4 v 416 4414 a(\(i\))48 b(Immediate.)389 4616 y(\(ii\))f(Giv)m(en)31 b(op)s(en)h Fo(G)c Fm(\022)g Fo(X)8 b Fq(,)31 b(then)i Fo(G)e Fq(is)g(a)h(union)f(of)g(basic)g(op)s (en)h(sets)h Fm(f)p Fo(B)3121 4631 y Fj(k)3191 4616 y Fq(:)28 b Fo(k)j Fm(2)d Fo(K)7 b Fm(g)568 4736 y Fq(in)30 b Fo(X)8 b Fq(,)31 b(whence)i Fo(\031)1222 4751 y Fj(i)1250 4736 y Fq(\()p Fo(G)p Fq(\))e(is)g(a)f(union)h(of)f(op)s(en)h(subsets)i Fm(f)p Fo(B)2688 4700 y Fj(i)2683 4761 y(k)2753 4736 y Fq(:)28 b Fo(k)j Fm(2)d Fo(K)7 b Fm(g)31 b Fq(of)f Fo(X)3345 4751 y Fj(i)3404 4736 y Fq(and)568 4857 y(is)j(therefore)h (op)s(en.)46 b(\(The)34 b(notation)e(here)i(is)f(in)m(tended)h(to)f (con)m(v)m(ey)j(that)d Fo(B)3425 4821 y Fj(i)3420 4881 y(k)3496 4857 y Fq(is)568 4977 y(the)g('comp)s(onen)m(t)f(along)f(the)i (i-th)f(co)s(ordinate)f(axis')i(of)f(the)h(op)s(en)g(b)s(o)m(x)g Fo(B)3355 4992 y Fj(k)3397 4977 y Fq(.\))1894 5251 y(38)p eop %%Page: 39 40 39 39 bop 324 548 a Fk(Theorem)37 b(4.1)49 b Fp(A)c(map)f(into)g(a)g (pr)-5 b(o)g(duct)45 b(sp)-5 b(ac)g(e)43 b(is)i(c)-5 b(ontinuous)44 b(i\013)g(its)h(c)-5 b(omp)g(osite)324 668 y(with)34 b(e)-5 b(ach)35 b(pr)-5 b(oje)g(ction)34 b(is)g(c)-5 b(ontinuous.)324 897 y Fq(Pro)s(of)p 324 910 235 4 v 27 w(Since)29 b(the)g(pro)5 b(jections)28 b(are)h(con)m(tin)m(uous,)h(so)e(m)m(ust)h(b)s(e)f(their)g(comp)s (osites)g(with)324 1017 y(an)m(y)46 b(con)m(tin)m(uous)f(map.)81 b(T)-8 b(o)45 b(establish)f(the)i(con)m(v)m(erse,)51 b(\014rst)45 b(sho)m(w)i(that)d(if)g Fm(S)53 b Fq(is)45 b(a)324 1137 y(subbase)30 b(for)f(the)g(co)s(domain)e(\(target\))i(of)f (a)h(mapping)e Fo(f)11 b Fq(,)30 b(then)f Fo(f)40 b Fq(will)27 b(b)s(e)i(con)m(tin)m(uous)324 1258 y(pro)m(vided)35 b(that)f(the)h(preimage)e(of)g(ev)m(ery)k(mem)m(b)s(er)c(of)h Fm(S)42 b Fq(is)34 b(op)s(en;)i(no)m(w)f(use)g(the)g(fact)324 1378 y(that)d(the)h(op)s(en)g(cylinders)g(constitute)f(a)h(subbase)h (for)e(the)h(pro)s(duct)g(top)s(ology)-8 b(.)324 1499 y Fp(Worke)j(d)47 b(example)d Fq(Sho)m(w)j(that)e(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))31 b Fm(\002)h Fq(\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))46 b(is)f(homeomorphic)f(to)i(\()p Fo(Y)5 b(;)17 b Fm(S)7 b Fq(\))31 b Fm(\002)324 1619 y Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\).)324 1739 y(Solution)p 324 1752 353 4 v 324 1963 a(De\014ne)87 b Fo(f)39 b Fq(:)28 b Fo(X)h Fm(\002)23 b Fo(Y)49 b Fm(!)27 b Fo(Y)44 b Fm(\002)22 b Fo(X)680 2084 y(g)31 b Fq(:)d Fo(Y)43 b Fm(\002)23 b Fo(X)36 b Fm(!)27 b Fo(X)j Fm(\002)22 b Fo(Y)1596 1764 y Ff(9)1596 1839 y(>)1596 1864 y(=)1596 2013 y(>)1596 2038 y(;)1705 1963 y Fq(b)m(y)78 b Fo(f)11 b Fq(\()p Fo(x;)17 b(y)t Fq(\))26 b(=)i(\()p Fo(y)t(;)17 b(x)p Fq(\))1885 2084 y Fo(g)t Fq(\()p Fo(y)t(;)g(x)p Fq(\))26 b(=)i(\()p Fo(x;)17 b(y)t Fq(\))p Fo(:)2623 1963 y Fq(Clearly)34 b(these)j(are)f(one-)324 2197 y(one,)d(on)m(to)g(and)g(m)m(utually)f (in)m(v)m(erse.)46 b(It)33 b(will)d(su\016ce)35 b(to)e(sho)m(w)h(that)f (b)s(oth)f(are)h(con)m(tin-)324 2318 y(uous.)324 2438 y Fo(\031)379 2453 y Fh(1)448 2438 y Fm(\016)c Fo(f)57 b Fq(=)45 b Fo(\031)812 2379 y Fd(0)808 2463 y Fh(2)848 2438 y Fq(;)j Fo(\031)978 2453 y Fh(2)1048 2438 y Fm(\016)29 b Fo(f)56 b Fq(=)46 b Fo(\031)1412 2379 y Fd(0)1408 2463 y Fh(1)1448 2438 y Fq(.)75 b(No)m(w)44 b Fo(\031)1842 2379 y Fd(0)1838 2463 y Fj(i)1911 2438 y Fq(is)f(con)m(tin)m(uous)h (for)f Fo(i)j Fq(=)f(1)p Fo(;)17 b Fq(2)43 b(and)g(so)g Fo(f)54 b Fq(is)324 2559 y(con)m(tin)m(uous!)44 b(Similarly)-8 b(,)29 b Fo(g)35 b Fq(is)d(con)m(tin)m(uous.)324 2679 y Fp(Worke)-5 b(d)38 b(example)d Fq(Sho)m(w)i(that)f(the)g(pro)s(duct)h (of)e(in\014nitely)g(man)m(y)h(copies)h(of)e(\()p Fo(N)5 b(;)17 b Fm(D)s Fq(\))324 2799 y(is)32 b(not)g(lo)s(cally)e(compact.) 324 2920 y(Solution)p 324 2933 V 324 3040 a(W)-8 b(e)37 b(claim)c(that)k Fp(no)e Fq(p)s(oin)m(t)h(has)h(a)f(compact)g(neigh)m (b)s(ourho)s(o)s(d.)54 b(Supp)s(ose)37 b(otherwise;)324 3161 y(then)f(there)h(exists)f Fo(p)e Fm(2)f Fo(X)8 b Fq(,)37 b Fo(C)j Fm(\022)34 b Fo(X)44 b Fq(and)35 b Fo(G)f Fm(\022)f Fo(X)44 b Fq(with)36 b Fo(C)42 b Fq(compact,)37 b Fo(G)e Fq(op)s(en)h(and)324 3281 y Fo(p)f Fm(2)h Fo(G)f Fm(\022)h Fo(C)7 b Fq(.)56 b(Pic)m(k)38 b(an)f(op)s(en)g(b)s(o)m(x)h Fo(B)k Fq(suc)m(h)c(that)f Fo(p)e Fm(2)h Fo(B)k Fm(\022)c Fo(G)f Fm(\022)h Fo(C)7 b Fq(.)57 b Fo(B)42 b Fq(lo)s(oks)36 b(lik)m(e)324 3401 y Fm(\\)390 3365 y Fj(n)390 3426 y(j)t Fh(=1)517 3401 y Fo(\031)576 3360 y Fg(\000)p Fh(1)572 3424 y Fj(i)596 3434 y Fi(j)670 3401 y Fq(\()p Fo(G)785 3416 y Fj(i)809 3426 y Fi(j)846 3401 y Fq(\).)70 b(Cho)s(ose)42 b Fo(i)1362 3416 y Fj(n)p Fh(+1)1542 3401 y Fm(2)h Fo(I)36 b Fm(n)28 b(f)p Fo(i)1891 3416 y Fh(1)1931 3401 y Fo(;)17 b(i)2008 3416 y Fh(2)2047 3401 y Fo(;)g(:)g(:)g(:)f(;)h(i)2299 3416 y Fj(n)2346 3401 y Fm(g)p Fq(;)46 b(then)c Fo(\031)2755 3416 y Fj(i)2779 3425 y Fi(n)p Fe(+1)2903 3401 y Fq(\()p Fo(C)7 b Fq(\))41 b(is)g(compact)324 3522 y(\(since)33 b(compactness)g(is)g(preserv)m(ed)i(b)m(y)e(con)m(tin)m(uous)g(maps\).) 324 3642 y(Th)m(us,)55 b Fo(p)668 3657 y Fj(i)692 3666 y Fi(n)p Fe(+1)871 3642 y Fm(2)h Fo(\031)1048 3657 y Fj(i)1072 3666 y Fi(n)p Fe(+1)1196 3642 y Fq(\()p Fo(B)5 b Fq(\))56 b(=)g Fo(X)1620 3657 y Fj(i)1644 3666 y Fi(n)p Fe(+1)1823 3642 y Fm(\022)h Fo(\031)2012 3657 y Fj(i)2036 3666 y Fi(n)p Fe(+1)2160 3642 y Fq(\()p Fo(C)7 b Fq(\))55 b Fm(\022)i Fo(X)2583 3657 y Fj(i)2607 3666 y Fi(n)p Fe(+1)2786 3642 y Fq(=)f(\()p Fo(N)5 b(;)17 b Fm(D)s Fq(\).)92 b(Th)m(us,)324 3762 y Fo(\031)379 3777 y Fj(i)403 3786 y Fi(n)p Fe(+1)527 3762 y Fq(\()p Fo(C)7 b Fq(\))27 b(=)h(\()p Fo(N)5 b(;)17 b Fm(D)s Fq(\))g Fo(:)g(:)g(:)31 b Fq(whic)m(h)i(is)f Fp(not)g Fq(compact!)324 4095 y Fn(4.2)160 b(Pro)t(ducts)53 b(and)h(T)-13 b(op)t(ological)55 b(Prop)t(erties)324 4314 y Fq(The)29 b(top)s(ological)c(prop)s(erties)j (p)s(ossessed)j(b)m(y)e(a)f(pro)s(duct)h(dep)s(ends,)i(of)d(course,)i (on)e(the)324 4435 y(prop)s(erties)39 b(p)s(ossessed)k(b)m(y)d(the)h (individual)c(factors.)65 b(There)41 b(are)f(sev)m(eral)g(theorems)324 4555 y(whic)m(h)49 b(assert)g(that)f(certain)g(top)s(ological)d(prop)s (erties)j(are)g Fk(pro)s(ductiv)m(e)g Fq(i.e.)91 b(are)324 4675 y(p)s(ossessed)29 b(b)m(y)g(the)e(pro)s(duct)h(if)e(enjo)m(y)m(ed) j(b)m(y)f(eac)m(h)g(factor.)41 b(Sev)m(eral)28 b(of)f(these)h(theorems) 324 4796 y(are)k(giv)m(en)h(b)s(elo)m(w.)1894 5251 y(39)p eop %%Page: 40 41 40 40 bop 324 548 a Fl(4.2.1)136 b(Pro)t(ducts)44 b(and)h (Connectedness)324 733 y Fk(Theorem)37 b(4.2)49 b Fp(A)n(ny)35 b(pr)-5 b(o)g(duct)35 b(of)f(c)-5 b(onne)g(cte)g(d)34 b(sp)-5 b(ac)g(es)34 b(must)h(b)-5 b(e)35 b(c)-5 b(onne)g(cte)g(d.)324 921 y Fq(Pro)s(of)p 324 934 235 4 v 32 w(is)32 b(left)f(to)i(the)g (reader.)324 1204 y Fl(4.2.2)136 b(Pro)t(ducts)44 b(and)h(Compactness) 324 1389 y Fk(Theorem)37 b(4.3)g(\()p Fp(T)-7 b(ychono\013)10 b('s)33 b(the)-5 b(or)g(em\))48 b(A)n(ny)25 b(pr)-5 b(o)g(duct)25 b(of)g(c)-5 b(omp)g(act)24 b(sp)-5 b(ac)g(es)24 b(is)h(c)-5 b(om-)324 1509 y(p)g(act)35 b(i.e.)44 b(c)-5 b(omp)g(actness)33 b(is)i(pr)-5 b(o)g(ductive.)324 1697 y Fq(Pro)s(of)p 324 1710 V 42 w(It)42 b(su\016ces)j(to)d(pro)m(v)m(e)h(that)f(an)m(y)i (co)m(v)m(ering)e(of)g Fo(X)51 b Fq(b)m(y)43 b(op)s(en)g(cylinders)f (has)h(a)324 1818 y(\014nite)36 b(sub)s(co)m(v)m(er.)57 b(Supp)s(ose)37 b(not)f(and)h(let)e Fm(C)43 b Fq(b)s(e)36 b(a)g(family)e(of)i(op)s(en)h(cylinders)f(whic)m(h)324 1938 y(co)m(v)m(ers)e Fo(X)41 b Fq(but)32 b(for)g(whic)m(h)i(no)e (\014nite)g(sub)s(co)m(v)m(er)j(exists.)44 b(F)-8 b(or)32 b(eac)m(h)h Fo(i)28 b Fm(2)g Fo(I)8 b Fq(,)33 b(consider)1195 2123 y Fm(f)p Fo(G)1322 2138 y Fj(i)1346 2148 y Fi(j)1410 2123 y Fq(:)28 b Fo(G)1542 2138 y Fj(i)1566 2148 y Fi(j)1630 2123 y Fm(\022)g Fo(X)1816 2138 y Fj(i)1877 2123 y Fq(and)33 b Fo(\031)2126 2082 y Fg(\000)p Fh(1)2122 2148 y Fj(i)2220 2123 y Fq(\()p Fo(G)2335 2138 y Fj(i)2359 2148 y Fi(j)2396 2123 y Fq(\))27 b Fm(2)h(C)6 b(g)p Fo(:)324 2308 y Fq(This)40 b(cannot)g(co)m(v)m(er)h Fo(X)1221 2323 y Fj(i)1288 2308 y Fq(\(otherwise,)i Fo(X)1878 2323 y Fj(i)1906 2308 y Fq(,)e(b)s(eing)e(compact,)i(w)m(ould)f(b)s(e)g(co)m(v)m(ered)i(b)m(y) 324 2428 y(\014nitely)32 b(man)m(y)-8 b(,)32 b(sa)m(y)i Fo(X)1186 2443 y Fj(i)1242 2428 y Fq(=)27 b Fo(G)1422 2443 y Fj(i)1446 2452 y Fe(1)1507 2428 y Fm([)22 b Fo(G)1672 2443 y Fj(i)1696 2452 y Fe(2)1757 2428 y Fm([)h Fo(:)17 b(:)g(:)22 b Fm([)g Fo(G)2148 2443 y Fj(i)2172 2451 y Fi(n)2219 2428 y Fq(,)32 b(whence)846 2613 y Fo(X)j Fq(=)28 b Fo(\031)1125 2572 y Fg(\000)p Fh(1)1121 2637 y Fj(i)1219 2613 y Fq(\()p Fo(X)1338 2628 y Fj(i)1366 2613 y Fq(\))g(=)244 b Fo(\031)1811 2572 y Fg(\000)p Fh(1)1807 2637 y Fj(i)1905 2613 y Fq(\()p Fo(G)2020 2628 y Fj(i)2044 2637 y Fe(1)2105 2613 y Fm([)23 b Fo(:)17 b(:)g(:)k Fm([)i Fo(\031)2478 2572 y Fg(\000)p Fh(1)2474 2637 y Fj(i)2572 2613 y Fq(\()p Fo(G)2687 2628 y Fj(i)2711 2636 y Fi(n)2758 2613 y Fq(\))p Fo(:)1752 2673 y Ff(|)p 1789 2673 461 10 v 461 w({z)p 2324 2673 V 461 w(})1535 2780 y Fq(all)31 b(in)h Fm(C)6 b Fq(,)33 b(con)m(trary)g(to)f(the)h(c)m(hoice)g(of)f Fm(C)324 2956 y Fq(Select,)k(therefore,)h Fo(z)1121 2971 y Fj(i)1182 2956 y Fm(2)32 b Fo(X)1361 2971 y Fj(i)1413 2956 y Fm(n)24 b([f)33 b Fq(those)g Fo(G)1968 2971 y Fj(i)1992 2981 y Fi(j)2029 2956 y Fq('s)p Fm(g)p Fq(;)k(consider)e Fo(z)j Fq(=)32 b(\()p Fo(z)2864 2971 y Fj(i)2892 2956 y Fq(\))2930 2971 y Fj(i)p Fg(2)p Fj(I)3073 2956 y Fm(2)h Fo(X)8 b Fq(.)51 b(Since)324 3076 y Fm(C)44 b Fq(co)m(v)m(ered)39 b Fo(X)8 b Fq(,)39 b Fo(z)i Fm(2)70 b Fq(some)32 b Fo(C)43 b Fm(2)37 b(C)6 b Fq(.)59 b(No)m(w)38 b Fo(C)44 b Fq(=)36 b Fo(\031)2267 3035 y Fg(\000)p Fh(1)2263 3101 y Fj(k)2361 3076 y Fq(\()p Fo(G)2476 3091 y Fj(k)2519 3076 y Fq(\))h(for)g(some)h Fo(k)h Fm(2)e Fo(I)46 b Fq(and)37 b(so)324 3196 y Fo(\031)379 3211 y Fj(k)422 3196 y Fq(\()p Fo(z)t Fq(\))28 b(=)g Fo(z)724 3211 y Fj(k)794 3196 y Fm(2)g Fo(G)965 3211 y Fj(k)1008 3196 y Fq(,)33 b(con)m(tradicting)e(the)i(c)m(hoice)g(of)f (the)h Fo(z)2443 3211 y Fj(i)2472 3196 y Fq('s.)324 3317 y(T)-8 b(o)28 b(pro)m(v)m(e)h(the)f(ab)s(o)m(v)m(e)h(without)f (Alexander's)h(Subbase)g(Theorem)f(is)g(v)m(ery)h(di\016cult)e(in)324 3437 y(general,)f(but)f(it)f(is)h(fairly)e(simple)h(in)g(the)i(sp)s (ecial)e(case)i(where)g Fo(I)33 b Fq(is)25 b(\014nite.)41 b(Sev)m(eral)25 b(fur-)324 3557 y(ther)31 b(results)g(sho)m(w)h(that)e (v)-5 b(arious)30 b(top)s(ological)d(prop)s(erties)k(are)g('\014nitely) f(pro)s(ductiv)m(e')324 3678 y(in)i(this)g(sense.)324 3851 y Fk(Theorem)37 b(4.4)49 b Fp(If)37 b Fq(\()p Fo(X)1215 3866 y Fh(1)1254 3851 y Fo(;)17 b Fm(T)1352 3866 y Fh(1)1391 3851 y Fq(\))p Fp(,)38 b Fq(\()p Fo(X)1616 3866 y Fh(2)1655 3851 y Fo(;)17 b Fm(T)1753 3866 y Fh(2)1793 3851 y Fq(\))p Fp(,)38 b(.)15 b(.)g(.)g(,)37 b Fq(\()p Fo(X)2220 3866 y Fj(n)2267 3851 y Fo(;)17 b Fm(T)2365 3866 y Fj(n)2412 3851 y Fq(\))37 b Fp(ar)-5 b(e)37 b(\014nitely)g(many)g(se)-5 b(quen-)324 3971 y(tial)5 b(ly)35 b(c)-5 b(omp)g(act)34 b(sp)-5 b(ac)g(es,)34 b(then)g(their)h(pr)-5 b(o)g(duct)35 b(is)g(se)-5 b(quential)5 b(ly)34 b(c)-5 b(omp)g(act.)324 4145 y Fq(Pro)s(of)p 324 4158 235 4 v 324 4265 a(T)d(ak)m(e)50 b(an)m(y)f(sequence)i(\()p Fo(x)1289 4280 y Fj(n)1337 4265 y Fq(\))k Fm(2)g Fo(X)8 b Fq(.)91 b(The)50 b(sequence)h(\()p Fo(\031)2488 4280 y Fh(1)2528 4265 y Fq(\()p Fo(x)2621 4280 y Fj(n)2668 4265 y Fq(\)\))2744 4280 y Fj(n)p Fg(\025)p Fh(1)2930 4265 y Fq(in)d(sequen)m(tially)324 4385 y(compact)e Fo(X)809 4400 y Fh(1)895 4385 y Fq(has)h(a)g(con)m(v)m(ergen)m(t)i (subsequence)h Fo(\031)2301 4400 y Fh(1)2341 4385 y Fq(\()p Fo(x)2434 4400 y Fj(n)2477 4412 y Fi(k)2519 4385 y Fq(\))i Fm(!)g Fo(l)2790 4400 y Fh(1)2881 4385 y Fm(2)g Fo(X)3080 4400 y Fh(1)3120 4385 y Fq(.)86 b(The)47 b(se-)324 4506 y(quence)24 b(\()p Fo(\031)729 4521 y Fh(2)769 4506 y Fq(\()p Fo(x)862 4521 y Fj(n)905 4533 y Fi(k)947 4506 y Fq(\)\))1023 4521 y Fj(k)r Fg(\025)p Fh(1)1179 4506 y Fq(in)d(sequen)m(tially)i(compact)f Fo(X)2268 4521 y Fh(2)2330 4506 y Fq(has)h(a)f(con)m(v)m(ergen)m(t)j(subsequence)324 4626 y(\()p Fo(\031)417 4641 y Fh(2)456 4626 y Fq(\()p Fo(x)549 4641 y Fj(n)592 4653 y Fi(k)625 4669 y(j)666 4626 y Fq(\)\))742 4641 y Fj(j)t Fg(\025)p Fh(1)897 4626 y Fm(!)i Fo(l)1053 4641 y Fh(2)1120 4626 y Fm(2)h Fo(X)1295 4641 y Fh(2)1367 4626 y Fq(and)33 b Fo(\031)1612 4641 y Fh(1)1652 4626 y Fq(\()p Fo(x)1745 4641 y Fj(n)1788 4653 y Fi(k)1821 4669 y(j)1862 4626 y Fq(\))27 b Fm(!)g Fo(l)2083 4641 y Fh(1)2155 4626 y Fq(also.)324 4761 y(Do)k(this)g Fo(n)h Fq(times!)43 b(W)-8 b(e)32 b(get)g(a)f(subsequence)36 b(\()p Fo(y)2103 4776 y Fj(p)2142 4761 y Fq(\))2180 4776 y Fj(p)p Fg(\025)p Fh(1)2341 4761 y Fq(of)c(the)g(original)d(sequence) 34 b(suc)m(h)324 4882 y(that)27 b Fo(\031)585 4897 y Fj(i)614 4882 y Fq(\()p Fo(y)700 4897 y Fj(p)739 4882 y Fq(\))h Fm(!)f Fo(l)961 4897 y Fj(i)1017 4882 y Fq(for)g Fo(i)h Fq(=)g(1)p Fo(;)17 b Fq(2)p Fo(;)g(:)g(:)g(:)e(;)i(n)p Fq(.)42 b(It's)28 b(easy)h(to)f(c)m(hec)m(k)i(that)d Fo(y)2816 4897 y Fj(p)2883 4882 y Fm(!)h Fq(\()p Fo(l)3078 4897 y Fh(1)3117 4882 y Fo(;)17 b(l)3190 4897 y Fh(2)3229 4882 y Fo(;)g(:)g(:)g(:)f(;)h(l)3477 4897 y Fj(n)3524 4882 y Fq(\))324 5002 y(so)33 b(that)f Fo(X)40 b Fq(is)32 b(sequen)m(tially)h(compact,)f(as)h(required.)1894 5251 y(40)p eop %%Page: 41 42 41 41 bop 324 548 a Fk(Lemma)37 b(4.3)49 b Fp(`The)34 b(pr)-5 b(o)g(duct)35 b(of)g(subsp)-5 b(ac)g(es)33 b(is)i(a)g(subsp)-5 b(ac)g(e)34 b(of)g(the)h(pr)-5 b(o)g(duct.')324 748 y Fq(Pro)s(of)p 324 761 235 4 v 324 869 a(Let)35 b(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))32 b(=)925 802 y Ff(Q)1003 890 y Fj(i)p Fg(2)p Fj(I)1114 869 y Fq(\()p Fo(X)1233 884 y Fj(i)1261 869 y Fo(;)17 b Fm(T)1359 884 y Fj(i)1387 869 y Fq(\);)37 b(let)e Fm(;)d(\032)h Fo(Y)1882 884 y Fj(i)1942 869 y Fm(\022)g Fo(X)2133 884 y Fj(i)2197 869 y Fq(for)h(eac)m(h)j Fo(i)32 b Fm(2)h Fo(I)8 b Fq(.)52 b(There)37 b(app)s(ear)e(to)324 989 y(b)s(e)e(t)m(w)m(o)g(di\013eren)m (t)g(w)m(a)m(ys)h(to)e(top)s(ologise)1840 923 y Ff(Q)1935 989 y Fo(Y)1992 1004 y Fj(i)2019 989 y Fq(:)141 1190 y Fp(either)g Fq(\(i\))48 b(giv)m(e)32 b(it)g(the)h(subspace)h(top)s (ology)d(induced)i(b)m(y)2341 1123 y Ff(Q)2436 1190 y Fm(T)2490 1205 y Fj(i)265 1392 y Fp(or)g Fq(\(ii\))47 b(giv)m(e)32 b(it)g(the)h(pro)s(duct)g(of)f(all)e(the)j(individual)d (subspace)35 b(top)s(ologies)30 b(\()p Fm(T)3230 1407 y Fj(i)3258 1392 y Fq(\))3296 1407 y Fj(Y)3337 1417 y Fi(i)3368 1392 y Fq(.)324 1593 y(The)36 b(p)s(oin)m(t)f(is)g(that)g (these)i(top)s(ologies)d(coincide|if)f Fo(G)2423 1557 y Fg(\003)2423 1617 y Fj(i)2447 1626 y Fe(0)2521 1593 y Fq(is)i(op)s(en)h(in)f(\()p Fm(T)3070 1608 y Fj(i)3094 1617 y Fe(0)3133 1593 y Fq(\))3171 1608 y Fj(Y)3212 1618 y Fi(i)3234 1633 y Fe(0)3312 1593 y Fq(where)324 1713 y Fo(i)357 1728 y Fh(0)424 1713 y Fm(2)28 b Fo(I)40 b Fq(i.e.)j Fo(G)846 1677 y Fg(\003)846 1738 y Fj(i)870 1747 y Fe(0)936 1713 y Fq(=)28 b Fo(Y)1097 1728 y Fj(i)1121 1737 y Fe(0)1180 1713 y Fm(\\)21 b Fo(G)1344 1728 y Fj(i)1368 1737 y Fe(0)1439 1713 y Fq(for)31 b(some)h Fo(G)1908 1728 y Fj(i)1932 1737 y Fe(0)1998 1713 y Fm(2)c(T)2146 1728 y Fj(i)2170 1737 y Fe(0)2209 1713 y Fq(,)k(a)g(t)m(ypical)f (subbasic)h(op)s(en)h(set)f(for)324 1834 y(\(ii\))e(is)1425 1954 y Fm(f)p Fq(\()p Fo(y)1561 1969 y Fj(i)1589 1954 y Fq(\))d Fm(2)1749 1871 y Ff(Y)1871 1954 y Fo(Y)1928 1969 y Fj(i)1984 1954 y Fq(:)g Fo(y)2086 1969 y Fj(i)2110 1978 y Fe(0)2176 1954 y Fm(2)i Fo(G)2348 1913 y Fg(\003)2348 1979 y Fj(i)2372 1988 y Fe(0)2410 1954 y Fm(g)324 2127 y Fq(whic)m(h)k(equals)982 2260 y Ff(Y)1105 2343 y Fo(Y)1162 2358 y Fj(i)1212 2343 y Fm(\\)22 b(f)p Fq(\()p Fo(x)1443 2358 y Fj(i)1472 2343 y Fq(\))27 b Fm(2)1632 2260 y Ff(Y)1754 2343 y Fo(X)1835 2358 y Fj(i)1891 2343 y Fq(:)h Fo(x)2001 2358 y Fj(i)2025 2367 y Fe(0)2092 2343 y Fm(2)g Fo(G)2263 2358 y Fj(i)2287 2367 y Fe(0)2353 2343 y Fm(2)g(T)2501 2358 y Fj(i)2525 2367 y Fe(0)2564 2343 y Fo(;)17 b(i)2641 2358 y Fh(0)2708 2343 y Fm(2)28 b Fo(I)8 b Fm(g)981 2581 y Fq(=)1084 2498 y Ff(Y)1207 2581 y Fo(Y)1264 2596 y Fj(i)1314 2581 y Fm(\\)22 b(f)33 b Fq(a)f(t)m(ypical)g(op)s(en)g (cylinder)h(in)2623 2498 y Ff(Y)2746 2581 y Fo(X)2827 2596 y Fj(i)2855 2581 y Fm(g)324 2754 y Fq(whic)m(h)g(is)f(a)g(t)m (ypical)g(subbasic)h(op)s(en)g(set)g(in)f(\(i\).)43 b(Hence,)34 b(\(i\))d(=)h(\(ii\).)324 2954 y Fk(Theorem)37 b(4.5)49 b Fp(L)-5 b(o)g(c)g(al)34 b(c)-5 b(omp)g(actness)34 b(is)g Fq(\014nitely)h Fp(pr)-5 b(o)g(ductive.)324 3155 y Fq(Pro)s(of)p 324 3168 V 324 3275 a(Giv)m(en)41 b Fo(x)i Fq(=)f(\()p Fo(x)924 3290 y Fh(1)964 3275 y Fo(;)17 b(x)1063 3290 y Fh(2)1102 3275 y Fo(;)g(:)g(:)g(:)f(;)h(x)1376 3290 y Fj(n)1423 3275 y Fq(\))42 b Fm(2)h Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))42 b(=)2055 3209 y Ff(Q)2134 3235 y Fj(n)2134 3300 y(i)p Fh(=1)2252 3275 y Fq(\()p Fo(X)2371 3290 y Fj(i)2399 3275 y Fo(;)17 b Fm(T)2497 3290 y Fj(i)2525 3275 y Fq(\),)43 b(w)m(e)g(m)m(ust)e(sho)m(w)h(that)f Fo(x)324 3396 y Fq(has)c(a)g(compact)g(neigh)m(b)s(ourho)s(o)s(d.)56 b(No)m(w,)38 b(for)f(all)e Fo(i)g Fq(=)g(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(n)p Fq(,)38 b Fo(x)2874 3411 y Fj(i)2940 3396 y Fq(has)f(a)g(compact)324 3516 y(neigh)m(b)s(ourho)s(o)s(d)25 b Fo(C)1054 3531 y Fj(i)1108 3516 y Fq(in)g(\()p Fo(X)1334 3531 y Fj(i)1362 3516 y Fo(;)17 b Fm(T)1460 3531 y Fj(i)1489 3516 y Fq(\))26 b(so)g(w)m(e)h(c)m(ho)s(ose)g Fm(T)2160 3531 y Fj(i)2189 3516 y Fq(-op)s(en)e(set)i Fo(G)2673 3531 y Fj(i)2727 3516 y Fq(suc)m(h)h(that)e Fo(x)3201 3531 y Fj(i)3257 3516 y Fm(2)i Fo(G)3428 3531 y Fj(i)3484 3516 y Fm(\022)324 3636 y Fo(C)394 3651 y Fj(i)422 3636 y Fq(.)43 b(Then)860 3853 y Fo(x)28 b Fm(2)g Fo(G)1114 3868 y Fh(1)1176 3853 y Fm(\002)22 b Fo(G)1352 3868 y Fh(2)1414 3853 y Fm(\002)g Fo(:)17 b(:)g(:)22 b Fm(\002)h Fo(G)1827 3868 y Fj(n)1037 3903 y Ff(|)p 1074 3903 344 10 v 344 w({z)p 1492 3903 V 344 w(})1280 4003 y Fg(\\)1327 3981 y Fi(n)1327 4026 y Fe(1)1369 4003 y Fj(\031)1412 3974 y Fd(\000)p Fe(1)1410 4027 y Fi(i)1495 4003 y Fh(\()p Fj(G)1577 4013 y Fi(i)1603 4003 y Fh(\))1901 3853 y Fm(\022)130 b Fo(C)2178 3868 y Fh(1)2240 3853 y Fm(\002)22 b Fo(C)2409 3868 y Fh(2)2471 3853 y Fm(\002)g Fo(:)17 b(:)g(:)22 b Fm(\002)h Fo(C)2877 3868 y Fj(n)2108 3903 y Ff(|)p 2145 3903 333 10 v 333 w({z)p 2552 3903 V 333 w(})2039 4011 y Fq(compact)32 b(subset)i(of)2851 3953 y Ff(Q)2941 4011 y Fj(X)2999 4021 y Fi(i)324 4223 y Fq(i.e.)50 b Fo(x)36 b Fq(has)f Fo(C)836 4238 y Fh(1)899 4223 y Fm(\002)24 b Fo(C)1070 4238 y Fh(2)1133 4223 y Fm(\002)h Fo(:)17 b(:)g(:)23 b Fm(\002)h Fo(C)1544 4238 y Fj(n)1626 4223 y Fq(as)35 b(a)g(compact)f(neigh)m(b)s(ourho)s(o)s(d.)50 b(\(Note)35 b(that)g(the)324 4344 y(previous)41 b(lemma)d(is)i(used)i (here,)i(to)c(allo)m(w)f(us)i(to)f(apply)h(T)m(yc)m(hono\013)7 b('s)43 b(theorem)d(to)324 4464 y(the)34 b(pro)s(duct)h(of)e(the)i (compact)e(subspaces)k Fo(C)2052 4479 y Fj(i)2080 4464 y Fq(,)d(and)g(then)h(to)f(view)g(this)g(ob)5 b(ject)35 b(as)f(a)324 4584 y(subspace)g(of)e(the)h(full)e(pro)s(duct!\))44 b(Th)m(us,)34 b Fo(X)40 b Fq(is)32 b(lo)s(cally)e(compact.)324 4802 y Fk(Lemma)37 b(4.4)p 913 4724 180 4 v 913 4735 a Ff(Q)1008 4802 y Fo(Y)1065 4817 y Fj(i)1093 4743 y Fg(T)1181 4802 y Fq(=)1284 4735 y Ff(Q)1397 4776 y Fq(\026)1379 4802 y Fo(Y)1436 4817 y Fj(i)1464 4746 y Fg(T)1503 4756 y Fi(i)1568 4802 y Fp(\()e(in)g(notation)f(of)h(pr)-5 b(evious)34 b(lemma\).)324 5002 y Fq(Pro)s(of)p 324 5015 235 4 v 25 w(Do)26 b(it)f(y)m(ourself)7 b(!)41 b(\(`The)28 b(closure)e(of)f(a)h(pro)s(duct)h(is)e(a)h(pro)s(duct)h(of)e(the)i (closures.'\))1894 5251 y(41)p eop %%Page: 42 43 42 42 bop 324 548 a Fl(4.2.3)136 b(Pro)t(ducts)44 b(and)h(Separabilit)l (y)324 733 y Fk(Theorem)37 b(4.6)49 b Fp(Sep)-5 b(ar)g(ability)34 b(is)h Fq(\014nitely)f Fp(pr)-5 b(o)g(ductive.)324 961 y Fq(Pro)s(of)p 324 974 235 4 v 324 1081 a(F)d(or)45 b(1)51 b Fm(\024)h Fo(i)g Fm(\024)g Fo(n)p Fq(,)e(c)m(ho)s(ose)d(coun)m (table)g Fo(D)1950 1096 y Fj(i)2029 1081 y Fm(\022)52 b Fo(X)2239 1096 y Fj(i)2313 1081 y Fq(where)2639 1056 y(\026)2609 1081 y Fo(D)2690 1096 y Fj(i)2718 1026 y Fg(T)2757 1036 y Fi(i)2839 1081 y Fq(=)f Fo(X)3047 1096 y Fj(i)3075 1081 y Fq(.)85 b(Consider)324 1213 y Fo(D)42 b Fq(=)d Fo(D)643 1228 y Fh(1)710 1213 y Fm(\002)27 b Fo(D)895 1228 y Fh(2)961 1213 y Fm(\002)h Fo(:)17 b(:)g(:)26 b Fm(\002)h Fo(D)1392 1228 y Fj(n)1479 1213 y Fq(=)1594 1146 y Ff(Q)1673 1173 y Fj(n)1673 1237 y Fh(1)1736 1213 y Fo(D)1817 1228 y Fj(i)1846 1213 y Fq(,)41 b(again)d(coun)m(table.)64 b(Then)2966 1188 y(\026)2943 1213 y Fo(D)42 b Fq(=)p 3182 1135 205 4 v 3182 1146 a Ff(Q)3277 1213 y Fo(D)3358 1228 y Fj(i)3386 1154 y Fg(T)3486 1213 y Fq(=)324 1280 y Ff(Q)449 1321 y Fq(\026)419 1346 y Fo(D)500 1361 y Fj(i)528 1291 y Fg(T)567 1301 y Fi(i)625 1346 y Fq(=)729 1280 y Ff(Q)824 1346 y Fo(X)905 1361 y Fj(i)960 1346 y Fq(=)28 b Fo(X)8 b Fq(.)324 1467 y(Notice)32 b(that)g(the)h(con)m(v)m (erses)j(of)c(all)e(suc)m(h)k(theorems)f(are)g(easily)f(true.)44 b(F)-8 b(or)31 b(example,)324 1670 y Fk(Theorem)37 b(4.7)49 b Fp(If)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))h(=)1507 1604 y Ff(Q)1585 1691 y Fj(i)p Fg(2)p Fj(I)1696 1670 y Fq(\()p Fo(X)1815 1685 y Fj(i)1843 1670 y Fo(;)17 b Fm(T)1941 1685 y Fj(i)1970 1670 y Fq(\))34 b Fp(is)409 1873 y(\(i\))49 b(c)-5 b(omp)g(act)379 2077 y(\(ii\))49 b(se)-5 b(quential)5 b(ly)34 b(c)-5 b(omp)g(act)350 2280 y(\(iii\))48 b(lo)-5 b(c)g(al)5 b(ly)34 b(c)-5 b(omp)g(act)364 2484 y(\(iv\))49 b(c)-5 b(onne)g(cte)g(d)394 2687 y(\(v\))49 b(sep)-5 b(ar)g(able)364 2891 y(\(vi\))49 b(c)-5 b(ompletely)34 b(sep)-5 b(ar)g(able)324 3094 y(then)34 b(so)h(is)g(every)f(`factor)h (sp)-5 b(ac)g(e')34 b Fq(\()p Fo(X)1743 3109 y Fj(i)1771 3094 y Fo(;)17 b Fm(T)1869 3109 y Fj(i)1897 3094 y Fq(\))p Fp(.)324 3297 y Fq(Pro)s(of)p 324 3310 235 4 v 39 w(F)-8 b(or)39 b(eac)m(h)i Fo(i)g Fm(2)g Fo(I)8 b Fq(,)42 b(the)f(pro)5 b(jection)39 b(mapping)g Fo(\031)2418 3312 y Fj(i)2487 3297 y Fq(:)i Fo(X)48 b Fm(!)40 b Fo(X)2905 3312 y Fj(i)2973 3297 y Fq(is)g(con)m(tin)m(uous,)324 3418 y(op)s(en)33 b(and)f(on)m(to.)44 b(Th)m(us,)34 b(b)m(y)f(previous)g(results,)g(the)g (result)g(follo)m(ws.)1894 5251 y(42)p eop %%Page: 43 44 43 43 bop 324 1212 a Fr(Chapter)78 b(5)324 1627 y(Separation)g(Axioms) 324 2080 y Fq(W)-8 b(e)37 b(ha)m(v)m(e)h(observ)m(ed)h(instances)f(of)f (top)s(ological)c(statemen)m(ts)38 b(whic)m(h,)g(although)e(true)324 2200 y(for)30 b(all)e(metric)i(\(and)g(metrizable\))f(spaces,)k(fail)28 b(for)i(some)g(other)h(top)s(ological)c(spaces.)324 2320 y(F)-8 b(requen)m(tly)g(,)48 b(the)d(cause)h(of)e(failure)f(can)h(b)s (e)h(traced)g(to)g(there)g(b)s(eing)f(`not)g(enough)324 2441 y(op)s(en)32 b(sets')i(\(in)d(senses)k(to)d(b)s(e)g(made)g (precise\).)44 b(F)-8 b(or)32 b(instance,)h(in)e(an)m(y)i(metric)e (space,)324 2561 y(compact)40 b(subsets)j(are)e(alw)m(a)m(ys)g(closed;) 46 b(but)41 b(not)f(in)g(ev)m(ery)j(top)s(ological)37 b(space,)44 b(for)324 2682 y(the)33 b(pro)s(of)f(ultimately)d(dep)s (ends)34 b(on)f(the)g(observ)-5 b(ation)568 2881 y(`giv)m(en)41 b Fo(x)h Fm(6)p Fq(=)g Fo(y)t Fq(,)h(it)d(is)g(p)s(ossible)h(to)f (\014nd)i(disjoin)m(t)e(op)s(en)h(sets)h Fo(G)f Fq(and)g Fo(H)568 3001 y Fq(with)32 b Fo(x)c Fm(2)g Fo(G)33 b Fq(and)f Fo(y)f Fm(2)d Fo(H)8 b Fq(')324 3201 y(whic)m(h)41 b(is)f(true)h(in)f(a)h(metric)f(space)h(\(e.g.)g(put)g Fo(G)h Fq(=)f Fo(B)5 b Fq(\()p Fo(x;)17 b(\017)p Fq(\),)44 b Fo(H)49 b Fq(=)41 b Fo(B)5 b Fq(\()p Fo(y)t(;)17 b(\017)p Fq(\))40 b(where)324 3321 y Fo(\017)28 b Fq(=)504 3282 y Fh(1)p 504 3298 36 4 v 504 3356 a(2)550 3321 y Fo(d)p Fq(\()p Fo(x;)17 b(y)t Fq(\)\))31 b(but)i(fails)e(in,)g(for)h(example,) h(a)f(trivial)e(space)k(\()p Fo(X)r(;)17 b Fm(T)2830 3336 y Fh(0)2869 3321 y Fq(\).)324 3442 y(What)29 b(w)m(e)i(do)e(no)m (w)h(is)f(to)g(see)i(ho)m(w)f(`demanding)e(certain)h(minim)m(um)d(lev)m (els-of-supply)324 3562 y(of)36 b(op)s(en)h(sets')h(gradually)d (eliminates)f(the)j(more)f(pathological)e(top)s(ologies,)i(lea)m(ving) 324 3682 y(us)d(with)f(those)h(whic)m(h)h(b)s(eha)m(v)m(e)g(lik)m(e)e (metric)f(spaces)j(to)e(a)h(greater)f(or)h(lesser)g(exten)m(t.)324 4015 y Fn(5.1)160 b Fc(T)773 4036 y Fq(1)880 4015 y Fn(Spaces)324 4234 y Fk(De\014nition)36 b(5.1)49 b Fp(A)36 b(top)-5 b(olo)g(gic)g(al)35 b(sp)-5 b(ac)g(e)35 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))36 b Fp(is)f Fo(T)2355 4249 y Fh(1)2431 4234 y Fp(if,)h(for)g(e)-5 b(ach)35 b Fo(x)h Fp(in)g Fo(X)8 b Fp(,)36 b Fm(f)p Fo(x)p Fm(g)g Fp(is)324 4354 y(close)-5 b(d.)324 4577 y Fk(Commen)m(t)36 b(5.1)134 b Fp(\(i\))49 b(Every)35 b(metrizable)f(sp)-5 b(ac)g(e)34 b(is)g Fo(T)2436 4592 y Fh(1)379 4779 y Fp(\(ii\))49 b Fq(\()p Fo(X)r(;)17 b Fm(T)787 4794 y Fh(0)826 4779 y Fq(\))35 b Fp(isn)-10 b('t)35 b Fo(T)1168 4794 y Fh(1)1242 4779 y Fp(unless)g Fm(j)p Fo(X)8 b Fm(j)27 b Fq(=)g(1)324 5002 y Fk(Theorem)37 b(5.1)134 b Fp(\(i\))49 b Fo(T)1292 5017 y Fh(1)1366 5002 y Fp(is)35 b(her)-5 b(e)g(ditary)1894 5251 y Fq(43)p eop %%Page: 44 45 44 44 bop 379 548 a Fp(\(ii\))49 b Fo(T)625 563 y Fh(1)699 548 y Fp(is)35 b(pr)-5 b(o)g(ductive)350 750 y(\(iii\))48 b Fo(T)625 765 y Fh(1)692 750 y Fm(\))34 b Fp(every)g(\014nite)g(set)g (is)g(close)-5 b(d.)43 b(Mor)-5 b(e)34 b(pr)-5 b(e)g(cisely,)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))34 b Fp(is)g Fo(T)3064 765 y Fh(1)3137 750 y Fp(i\013)g Fm(T)53 b(\023)28 b(C)6 b Fp(,)568 871 y(i.e.)34 b Fm(C)41 b Fp(is)35 b(the)g(we)-5 b(akest)34 b(of)g(al)5 b(l)35 b(the)g Fo(T)1921 886 y Fh(1)1995 871 y Fp(top)-5 b(olo)g(gies)34 b(that)h(c)-5 b(an)35 b(b)-5 b(e)34 b(de\014ne)-5 b(d)34 b(on)h Fo(X)8 b Fp(.)324 1096 y Fq(Pro)s(of)p 324 1109 235 4 v 32 w(is)32 b(left)f(to)i(the)g(reader.)324 1216 y(The)d(resp)s(ects)h(in)e(whic)m (h)h Fo(T)1334 1231 y Fh(1)1373 1216 y Fq(-spaces)h(are)e(`nicer')h (than)f(others)h(are)g(mostly)e(concerned)324 1336 y(with)k(`cluster)h (p)s(oin)m(t)f(of)h(a)f(set')i(\(an)e(idea)h(w)m(e)g(ha)m(v)m(e)i(a)m (v)m(oided!\).)44 b(W)-8 b(e)33 b(sho)m(w)h(the)g(equiv-)324 1457 y(alence,)f(in)e Fo(T)814 1472 y Fh(1)886 1457 y Fq(spaces,)k(of)d(the)h(t)m(w)m(o)g(forms)f(of)g(its)g(de\014nition)g (used)h(in)f(analysis.)324 1682 y Fk(Theorem)37 b(5.2)49 b Fp(Given)39 b(a)h Fo(T)1427 1697 y Fh(1)1507 1682 y Fp(sp)-5 b(ac)g(e)39 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))p Fp(,)41 b Fo(p)c Fm(2)h Fo(X)47 b Fp(and)40 b Fo(A)d Fm(\022)g Fo(X)8 b Fp(,)41 b(the)f(fol)5 b(lowing)324 1802 y(ar)-5 b(e)34 b(e)-5 b(quivalent:)409 2003 y(\(i\))49 b(Every)35 b(neighb)-5 b(ourho)g(o)g(d)33 b(of)i Fo(p)f Fp(c)-5 b(ontains)35 b(in\014nitely)f(many)h(p)-5 b(oints)34 b(of)h Fo(A)379 2205 y Fp(\(ii\))49 b(Every)28 b(neighb)-5 b(ourho)g(o)g(d)27 b(of)i Fo(p)f Fp(c)-5 b(ontains)28 b(at)h(le)-5 b(ast)28 b(one)g(p)-5 b(oint)28 b(of)g Fo(A)h Fp(di\013er)-5 b(ent)28 b(fr)-5 b(om)568 2326 y Fo(p)p Fp(.)324 2551 y Fq(Pro)s(of)p 324 2564 V 41 w(Ob)m(viously)d(,)45 b(\(i\))c Fm(\))h Fq(\(ii\);)j(con)m(v)m(ersely)-8 b(,)47 b(supp)s(ose)c(\(i\))e(fails;)46 b(so)c(there)h(exists)g(a)324 2671 y(neigh)m(b)s(ourho)s(o)s(d)36 b Fo(N)47 b Fq(of)37 b Fo(p)g Fq(suc)m(h)h(that)f Fo(N)f Fm(\\)26 b Fo(A)37 b Fq(is)f(\014nite.)57 b(Consider)37 b Fo(H)43 b Fq(=)35 b([)p Fo(X)f Fm(n)25 b Fq(\()p Fo(N)35 b Fm(\\)324 2791 y Fo(A)p Fq(\)])t Fm([)t(f)p Fo(p)p Fm(g)p Fq(;)26 b(it)d(is)g (co\014nite)h(and)f(is)g(th)m(us)i(an)f(\(op)s(en\))f(neigh)m(b)s (ourho)s(o)s(d)g(of)g Fo(p)p Fq(.)41 b(Hence)24 b Fo(N)14 b Fm(\\)t Fo(H)324 2912 y Fq(is)23 b(a)g(neigh)m(b)s(ourho)s(o)s(d)f (of)h Fo(p)g Fq(whic)m(h)h(con)m(tains)f(no)g(p)s(oin)m(ts)g(of)g Fo(A)p Fq(,)i(except)g(p)s(ossibly)e Fo(p)g Fq(itself.)324 3032 y(Th)m(us,)34 b(\(ii\))d(fails)f(also.)324 3153 y(Hence,)k(\(i\))d Fm(,)h Fq(\(ii\).)324 3485 y Fn(5.2)160 b Fc(T)773 3506 y Fq(2)880 3485 y Fn(\(Hausdor\013)15 b(\))53 b(Spaces)324 3704 y Fk(De\014nition)36 b(5.2)49 b Fp(A)43 b(top)-5 b(olo)g(gic)g(al)43 b(sp)-5 b(ac)g(e)42 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))43 b Fp(is)g Fo(T)2392 3719 y Fh(2)2474 3704 y Fp(\(or)g Fk(Hausdor\013)p Fp(\))i(i\013)d(given)324 3824 y Fo(x)28 b Fm(6)p Fq(=)f Fo(y)38 b Fp(in)d Fo(X)8 b Fp(,)35 b Fm(9)g Fp(disjoint)f(neighb)-5 b(ourho)g(o)g(ds)33 b(of)i Fo(x)g Fp(and)f Fo(y)t Fp(.)324 4049 y Fk(Commen)m(t)i(5.2)134 b Fp(\(i\))49 b(Every)35 b(metrizable)f(sp)-5 b(ac)g(e)34 b(is)g Fo(T)2436 4064 y Fh(2)379 4252 y Fp(\(ii\))49 b Fo(T)625 4267 y Fh(2)695 4252 y Fm(\))30 b Fo(T)882 4267 y Fh(1)958 4252 y Fp(\(i.e.)35 b(any)h Fo(T)1413 4267 y Fh(2)1489 4252 y Fp(sp)-5 b(ac)g(e)36 b(is)g Fo(T)1908 4267 y Fh(1)1947 4252 y Fp(,)h(for)f(if)g Fo(x)p Fp(,)h Fo(y)d Fm(2)d Fo(T)2625 4267 y Fh(2)2664 4252 y Fo(X)44 b Fp(and)36 b Fo(y)d Fm(2)p 3158 4167 156 4 v 31 w(f)p Fo(x)p Fm(g)p Fp(,)k(then)568 4372 y(every)d(neighb)-5 b(ourho)g(o)g(d)34 b(of)g Fo(y)k Fp(c)-5 b(ontains)34 b Fo(x)p Fp(,)i(whenc)-5 b(e)33 b Fo(x)c Fq(=)e Fo(y)t Fp(.\))350 4575 y(\(iii\))48 b Fq(\()p Fo(X)r(;)17 b Fm(C)6 b Fq(\))p Fp(,)35 b(with)f Fo(X)43 b Fp(in\014nite,)34 b(c)-5 b(annot)34 b(b)-5 b(e)35 b Fo(T)2088 4590 y Fh(2)324 4800 y Fk(Theorem)i(5.3)134 b Fp(\(i\))49 b Fo(T)1292 4815 y Fh(2)1366 4800 y Fp(is)35 b(her)-5 b(e)g(ditary)379 5002 y(\(ii\))49 b Fo(T)625 5017 y Fh(2)699 5002 y Fp(is)35 b(pr)-5 b(o)g(ductive.)1894 5251 y Fq(44)p eop %%Page: 45 46 45 45 bop 324 548 a Fq(Pro)s(of)p 324 561 235 4 v 492 751 a(i)48 b(The)33 b(pro)s(of)f(is)g(left)g(to)g(the)h(reader.)465 955 y(ii)47 b(Let)30 b(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))h(=)1154 888 y Ff(Q)1232 976 y Fj(i)p Fg(2)p Fj(I)1343 955 y Fq(\()p Fo(X)1462 970 y Fj(i)1490 955 y Fo(;)17 b Fm(T)1588 970 y Fj(i)1616 955 y Fq(\))31 b(b)s(e)f(an)m(y)h(pro)s(duct)f(of)g Fo(T)2526 970 y Fh(2)2596 955 y Fq(spaces.)44 b(Let)31 b Fo(x)d Fq(=)f(\()p Fo(x)3384 970 y Fj(i)3413 955 y Fq(\))3451 970 y Fj(i)p Fg(2)p Fj(I)568 1075 y Fq(and)40 b Fo(y)45 b Fq(=)d(\()p Fo(y)1062 1090 y Fj(i)1089 1075 y Fq(\))1127 1090 y Fj(i)p Fg(2)p Fj(I)1279 1075 y Fq(b)s(e)f(distinct) f(elemen)m(ts)h(of)f Fo(X)8 b Fq(.)68 b(Then)42 b(there)f(exists)h Fo(i)3322 1090 y Fh(0)3403 1075 y Fm(2)g Fo(I)568 1196 y Fq(suc)m(h)32 b(that)e Fo(x)1050 1211 y Fj(i)1074 1220 y Fe(0)1141 1196 y Fm(6)p Fq(=)d Fo(y)1292 1211 y Fj(i)1316 1220 y Fe(0)1385 1196 y Fq(in)j Fo(X)1578 1211 y Fj(i)1602 1220 y Fe(0)1641 1196 y Fq(.)43 b(Cho)s(ose)31 b(disjoin)m(t)e(op)s(en) i(sets)h Fo(G)p Fq(,)f Fo(H)38 b Fq(in)29 b(\()p Fo(X)3300 1211 y Fj(i)3324 1220 y Fe(0)3363 1196 y Fo(;)17 b Fm(T)3461 1211 y Fj(i)3485 1220 y Fe(0)3524 1196 y Fq(\))568 1316 y(so)33 b(that)g Fo(x)955 1331 y Fj(i)979 1340 y Fe(0)1046 1316 y Fm(2)c Fo(G)p Fq(,)34 b Fo(y)1327 1331 y Fj(i)1351 1340 y Fe(0)1417 1316 y Fm(2)29 b Fo(H)8 b Fq(.)45 b(Then)34 b Fo(x)29 b Fm(2)g Fo(\031)2166 1275 y Fg(\000)p Fh(1)2162 1339 y Fj(i)2186 1348 y Fe(0)2260 1316 y Fq(\()p Fo(G)p Fq(\))f Fm(2)h(T)d Fq(,)33 b Fo(y)f Fm(2)d Fo(\031)2910 1275 y Fg(\000)p Fh(1)2906 1339 y Fj(i)2930 1348 y Fe(0)3004 1316 y Fq(\()p Fo(H)8 b Fq(\))28 b Fm(2)h(T)58 b Fq(and)568 1436 y(since)33 b Fo(G)22 b Fm(\\)g Fo(H)35 b Fq(=)28 b Fm(;)p Fq(,)k Fo(\031)1382 1395 y Fg(\000)p Fh(1)1378 1459 y Fj(i)1402 1468 y Fe(0)1477 1436 y Fq(\()p Fo(G)p Fq(\))22 b Fm(\\)g Fo(\031)1799 1395 y Fg(\000)p Fh(1)1795 1459 y Fj(i)1819 1468 y Fe(0)1893 1436 y Fq(\()p Fo(H)8 b Fq(\))28 b(=)f Fm(;)p Fq(.)43 b(Hence)34 b(result.)324 1640 y(The)i Fo(T)584 1655 y Fh(2)658 1640 y Fq(axiom)d(is)i (particularly)d(v)-5 b(aluable)34 b(when)i(exploring)d(compactness.)52 b(P)m(art)35 b(of)324 1760 y(the)c(reason)g(is)f(that)h Fo(T)1155 1775 y Fh(2)1225 1760 y Fq(implies)d(that)j(p)s(oin)m(ts)f (and)h(compact)f(sets)i(can)f(b)s(e)g(`separated)324 1880 y(o\013)7 b(')36 b(b)m(y)h(op)s(en)g(sets)h(and)e(ev)m(en)i (implies)c(that)i(compact)g(sets)i(can)f(b)s(e)f(`separated)i(o\013)7 b(')324 2001 y(from)31 b(other)i(compact)f(sets)i(in)e(the)h(same)f(w)m (a)m(y)-8 b(.)324 2204 y Fk(Theorem)37 b(5.4)49 b Fp(In)35 b(a)g Fo(T)1261 2219 y Fh(2)1301 2204 y Fp(-sp)-5 b(ac)g(e)35 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))p Fp(,)36 b(if)f Fo(C)43 b Fp(is)35 b(a)h(c)-5 b(omp)g(act)34 b(set)i(and)f Fo(x)30 b Fm(62)f Fo(C)7 b Fp(,)36 b(then)324 2325 y(ther)-5 b(e)35 b(exist)f Fm(T)26 b Fp(-op)-5 b(en)34 b(sets)h Fo(G)f Fp(and)h Fo(H)42 b Fp(so)35 b(that)g Fo(x)28 b Fm(2)g Fo(G)p Fp(,)35 b Fo(C)f Fm(\022)29 b Fo(H)42 b Fp(and)34 b Fo(G)22 b Fm(\\)h Fo(H)35 b Fq(=)28 b Fm(;)p Fo(:)324 2528 y Fq(Pro)s(of)p 324 2541 V 41 w(A)43 b(v)-5 b(aluable)41 b(exercise:)64 b(separate)43 b(eac)m(h)g(p)s(oin)m(t)f(of) f(C)i(from)e(x)i(using)f(disjoin)m(t)324 2648 y(op)s(en)35 b(sets,)h(note)f(that)g(the)g(op)s(en)g(neigh)m(b)s(ourho)s(o)s(ds)f (of)h(the)g(v)-5 b(arious)34 b(elemen)m(ts)h(of)f(C,)324 2769 y(th)m(us)e(obtained,)f(mak)m(e)g(up)h(an)f(op)s(en)g(co)m(v)m (ering)g(of)g(C,)h(reduce)g(it)e(to)h(a)g(\014nite)g(sub)s(co)m(v)m(er) 324 2889 y(b)m(y)i(app)s(ealing)e(to)h(compactness)i(.)16 b(.)g(.)324 3117 y Fk(Corollary)36 b(5.1)49 b Fp(In)34 b(a)h Fo(T)1283 3132 y Fh(2)1322 3117 y Fp(-sp)-5 b(ac)g(e,)34 b(any)h(c)-5 b(omp)g(act)34 b(set)h(is)f(close)-5 b(d.)324 3346 y Fk(Corollary)36 b(5.2)49 b Fp(In)37 b(a)g Fo(T)1288 3361 y Fh(2)1328 3346 y Fp(-sp)-5 b(ac)g(e,)37 b(if)g Fo(C)45 b Fp(and)37 b Fo(K)45 b Fp(ar)-5 b(e)37 b(non-empty)g(c)-5 b(omp)g(act)37 b(and)g(dis-)324 3466 y(joint,)d(then)h(ther)-5 b(e)35 b(exist)f(op)-5 b(en)35 b Fo(G)p Fp(,)f Fo(H)43 b Fp(such)34 b(that)i Fo(C)e Fm(\022)28 b Fo(G)p Fp(,)35 b Fo(K)g Fm(\022)28 b Fo(H)43 b Fp(and)34 b Fo(G)22 b Fm(\\)g Fo(H)36 b Fq(=)27 b Fm(;)p Fo(:)324 3694 y Fq(A)43 b(basic)g(formal)e(distinction)h(b)s(et)m(w)m(een)j(algebra)d(and)h (top)s(ology)f(is)h(that)g(although)324 3815 y(the)30 b(in)m(v)m(erse)i(of)d(a)h(one-one,)h(on)m(to)e(group)h(homomorphism)d ([etc!])44 b(is)29 b(automatically)e(a)324 3935 y(homomorphism)f (again,)i(the)h(in)m(v)m(erse)h(of)e(a)g(one-one,)i(on)m(to)e(con)m (tin)m(uous)h(map)f(can)h(fail)324 4056 y(to)d(b)s(e)i(con)m(tin)m (uous.)42 b(It)27 b(is)g(a)f(consequence)k(of)d(Corollary)e(5.2)i (that,)h(amongst)e(compact)324 4176 y Fo(T)381 4191 y Fh(2)453 4176 y Fq(spaces,)34 b(this)e(cannot)h(happ)s(en.)324 4379 y Fk(Theorem)k(5.5)49 b Fp(L)-5 b(et)33 b Fo(f)39 b Fq(:)27 b(\()p Fo(X)1417 4394 y Fh(1)1457 4379 y Fo(;)17 b Fm(T)1555 4394 y Fh(1)1594 4379 y Fq(\))28 b Fm(!)f Fq(\()p Fo(X)1906 4394 y Fh(2)1945 4379 y Fo(;)17 b Fm(T)2043 4394 y Fh(2)2083 4379 y Fq(\))33 b Fp(b)-5 b(e)33 b(one-one,)f(onto)g (and)h(c)-5 b(ontinuous,)324 4500 y(wher)g(e)34 b Fo(X)680 4515 y Fh(1)754 4500 y Fp(is)h(c)-5 b(omp)g(act)34 b(and)g Fo(X)1505 4515 y Fh(2)1580 4500 y Fp(is)g Fo(T)1741 4515 y Fh(2)1781 4500 y Fp(.)44 b(Then)35 b Fo(f)45 b Fp(is)35 b(a)f(home)-5 b(omorphism.)324 4703 y Fq(Pro)s(of)p 324 4716 V 34 w(It)35 b(su\016ces)h(to)f(pro)m(v)m(e)h(that)e Fo(f)45 b Fq(is)35 b(closed.)50 b(Giv)m(en)34 b(closed)h Fo(K)k Fm(\022)31 b Fo(X)3045 4718 y Fh(1)3085 4703 y Fq(,)k(then)g Fo(K)42 b Fq(is)324 4824 y(compact)32 b(whence)i Fo(f)11 b Fq(\()p Fo(K)c Fq(\))32 b(is)g(compact)g(and)g(so)h Fo(f)11 b Fq(\()p Fo(K)c Fq(\))32 b(is)g(closed.)43 b(Th)m(us)34 b Fo(f)43 b Fq(is)32 b(a)g(closed)324 4944 y(map.)1894 5251 y(45)p eop %%Page: 46 47 46 46 bop 324 548 a Fk(Theorem)37 b(5.6)49 b Fq(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))35 b Fp(is)g Fo(T)1470 563 y Fh(2)1544 548 y Fp(i\013)g(no)f(net)h(in)g Fo(X)42 b Fp(has)35 b(mor)-5 b(e)34 b(than)h(one)f(limit.)324 741 y Fq(Pro)s(of)p 324 754 235 4 v 94 918 a(\(i\))e Fm(\))g Fq(\(ii\):)47 b(Let)25 b Fo(x)j Fm(6)p Fq(=)g Fo(y)g Fq(in)c Fo(X)8 b Fq(;)28 b(b)m(y)e(h)m(yp)s(othesis,)i(there)e (exist)f(disjoin)m(t)g(neigh)m(b)s(ourho)s(o)s(ds)f Fo(U)36 b Fq(of)568 1038 y Fo(x)p Fq(,)c Fo(V)53 b Fq(of)31 b Fo(y)t Fq(.)42 b(Since)31 b(a)h(net)f(cannot)h(ev)m(en)m(tually)g(b)s (elong)e(to)h(eac)m(h)i(of)e(t)m(w)m(o)h(disjoin)m(t)568 1158 y(sets,)i(it)d(is)h(clear)g(that)h(no)f(net)h(in)f Fo(X)40 b Fq(can)33 b(con)m(v)m(erge)h(to)f Fp(b)-5 b(oth)32 b Fo(x)h Fq(and)g Fo(y)t Fq(.)94 1353 y(\(ii\))e Fm(\))h Fq(\(i\):)48 b(Supp)s(ose)36 b(that)f(\()p Fo(X)r(;)17 b Fm(T)25 b Fq(\))35 b(is)g(not)g(Hausdor\013)g(and)g(that)g Fo(x)e Fm(6)p Fq(=)f Fo(y)38 b Fq(are)d(p)s(oin)m(ts)g(in)f Fo(X)568 1473 y Fq(for)f(whic)m(h)h(ev)m(ery)h(neigh)m(b)s(ourho)s(o)s (d)d(of)h Fo(x)h Fq(in)m(tersects)h(ev)m(ery)h(neigh)m(b)s(ourho)s(o)s (d)c(of)568 1594 y Fo(y)t Fq(.)64 b(Let)40 b Fm(N)975 1609 y Fj(x)1058 1594 y Fq(\()p Fm(N)1178 1609 y Fj(y)1219 1594 y Fq(\))g(b)s(e)g(the)g(neigh)m(b)s(ourho)s(o)s(d)f(systems)i(at)f Fo(x)g Fq(\()p Fo(y)t Fq(\))f(resp)s(ectiv)m(ely)-8 b(.)568 1714 y(Then)34 b(b)s(oth)e Fm(N)1135 1729 y Fj(x)1212 1714 y Fq(and)h Fm(N)1484 1729 y Fj(y)1558 1714 y Fq(are)g(directed)g (b)m(y)h(rev)m(erse)h(inclusion.)43 b(W)-8 b(e)33 b(order)g(the)568 1834 y(Cartesian)f(pro)s(duct)h Fm(N)1455 1849 y Fj(x)1521 1834 y Fm(\002)22 b(N)1702 1849 y Fj(y)1776 1834 y Fq(b)m(y)34 b(agreeing)d(that)1127 2023 y(\()p Fo(U)1231 2038 y Fj(x)1275 2023 y Fo(;)17 b(U)1385 2038 y Fj(y)1426 2023 y Fq(\))28 b Fm(\025)g Fq(\()p Fo(V)1692 2038 y Fj(x)1736 2023 y Fo(;)17 b(V)1837 2038 y Fj(y)1878 2023 y Fq(\))27 b Fm(,)h Fo(U)2137 2038 y Fj(x)2208 2023 y Fm(\022)h Fo(V)2371 2038 y Fj(x)2447 2023 y Fq(and)k Fo(U)2703 2038 y Fj(y)2772 2023 y Fm(\022)28 b Fo(V)2934 2038 y Fj(y)2975 2023 y Fo(:)568 2212 y Fq(Eviden)m(tly)-8 b(,)40 b(this)e(order)h(is)f (directed.)62 b(F)-8 b(or)37 b(eac)m(h)j(\()p Fo(U)2538 2227 y Fj(x)2582 2212 y Fo(;)17 b(U)2692 2227 y Fj(y)2733 2212 y Fq(\))38 b Fm(2)g(N)2995 2227 y Fj(x)3065 2212 y Fm(\002)27 b(N)3251 2227 y Fj(y)3292 2212 y Fq(,)40 b Fo(U)3425 2227 y Fj(x)3495 2212 y Fm(\\)568 2333 y Fo(U)634 2348 y Fj(y)712 2333 y Fm(6)p Fq(=)c Fm(;)h Fq(and)h(hence)h(w)m(e)g(ma)m(y)e(select)h(a)f(p)s(oin)m(t)g Fo(z)2409 2348 y Fh(\()p Fj(U)2484 2356 y Fi(x)2524 2348 y Fj(;U)2592 2356 y Fi(y)2628 2348 y Fh(\))2696 2333 y Fm(2)g Fo(U)2865 2348 y Fj(x)2935 2333 y Fm(\\)26 b Fo(U)3093 2348 y Fj(y)3134 2333 y Fq(.)59 b(If)38 b Fo(W)3415 2348 y Fj(x)3496 2333 y Fq(is)568 2453 y Fp(any)g Fq(neigh)m(b)s(ourho) s(o)s(d)g(of)g Fo(x)p Fq(,)j Fo(W)1763 2468 y Fj(y)1843 2453 y Fp(any)d Fq(neigh)m(b)s(ourho)s(o)s(d)g(of)g Fo(y)k Fq(and)d(\()p Fo(U)3213 2468 y Fj(x)3257 2453 y Fo(;)17 b(U)3367 2468 y Fj(y)3408 2453 y Fq(\))38 b Fm(\025)568 2574 y Fq(\()p Fo(W)698 2589 y Fj(x)742 2574 y Fo(;)17 b(W)878 2589 y Fj(y)919 2574 y Fq(\),)32 b(then)1422 2694 y Fo(z)1467 2709 y Fh(\()p Fj(U)1542 2717 y Fi(x)1581 2709 y Fj(;U)1649 2717 y Fi(y)1686 2709 y Fh(\))1745 2694 y Fm(2)c Fo(U)1905 2709 y Fj(x)1971 2694 y Fm(\\)23 b Fo(U)2126 2709 y Fj(y)2195 2694 y Fm(\022)28 b Fo(W)2392 2709 y Fj(x)2459 2694 y Fm(\\)22 b Fo(W)2639 2709 y Fj(y)2681 2694 y Fo(:)568 2855 y Fq(That)40 b(is,)h(the)f(net)g Fm(f)p Fo(z)1398 2871 y Fh(\()p Fj(U)1473 2879 y Fi(x)1512 2871 y Fj(;U)1580 2879 y Fi(y)1617 2871 y Fh(\))1648 2855 y Fo(;)17 b Fq(\()p Fo(U)1796 2870 y Fj(x)1840 2855 y Fo(;)g(U)1950 2870 y Fj(y)1991 2855 y Fq(\))40 b Fm(2)g(N)2257 2870 y Fj(x)2328 2855 y Fm(\002)27 b(N)2514 2870 y Fj(y)2555 2855 y Fm(g)40 b Fq(ev)m(en)m(tually)g(b)s(elongs)f(to)568 2975 y(b)s(oth)32 b Fo(W)890 2990 y Fj(x)966 2975 y Fq(and)h Fo(W)1248 2990 y Fj(y)1322 2975 y Fq(and)g(consequen)m(tly)i(con)m(v)m (erges)f(to)f Fp(b)-5 b(oth)32 b Fo(x)h Fq(and)g Fo(y)t Fq(!)324 3168 y Fk(Corollary)j(5.3)49 b Fp(L)-5 b(et)38 b Fo(f)45 b Fq(:)33 b(\()p Fo(X)1457 3183 y Fh(1)1497 3168 y Fo(;)17 b Fm(T)1595 3183 y Fh(1)1634 3168 y Fq(\))34 b Fm(!)f Fq(\()p Fo(X)1958 3183 y Fh(2)1998 3168 y Fo(;)17 b Fm(T)2096 3183 y Fh(2)2135 3168 y Fq(\))p Fp(,)39 b Fo(g)e Fq(:)d(\()p Fo(X)2506 3183 y Fh(1)2545 3168 y Fo(;)17 b Fm(T)2643 3183 y Fh(1)2683 3168 y Fq(\))33 b Fm(!)h Fq(\()p Fo(X)3007 3183 y Fh(2)3046 3168 y Fo(;)17 b Fm(T)3144 3183 y Fh(2)3183 3168 y Fq(\))38 b Fp(b)-5 b(e)38 b(c)-5 b(on-)324 3289 y(tinuous)40 b(wher)-5 b(e)40 b Fo(X)1037 3304 y Fh(2)1116 3289 y Fp(is)g Fo(T)1283 3304 y Fh(2)1323 3289 y Fp(.)60 b(Then)39 b(their)h(`agr)-5 b(e)g(ement)40 b(set')g(is)f(close)-5 b(d)40 b(i.e.)f Fo(A)f Fq(=)f Fm(f)p Fo(x)h Fq(:)324 3409 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))27 b(=)h Fo(g)t Fq(\()p Fo(x)p Fq(\))p Fm(g)34 b Fp(is)h(close)-5 b(d.)324 3737 y Fn(5.3)160 b Fc(T)773 3759 y Fq(3)880 3737 y Fn(Spaces)324 3956 y Fk(De\014nition)36 b(5.3)49 b Fp(A)35 b(sp)-5 b(ac)g(e)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))35 b Fp(is)f(c)-5 b(al)5 b(le)-5 b(d)34 b Fo(T)2148 3971 y Fh(3)2223 3956 y Fp(or)g Fk(regular)h Fp(pr)-5 b(ovide)g(d)34 b(:-)409 4133 y(\(i\))49 b(it)35 b(is)f Fo(T)826 4148 y Fh(1)866 4133 y Fp(,)h(and)379 4328 y(\(ii\))49 b(given)44 b Fo(x)i Fm(62)81 b Fp(close)-5 b(d)34 b Fo(F)14 b Fp(,)47 b(ther)-5 b(e)45 b(exist)f(disjoint)g(op)-5 b(en)44 b(sets)h Fo(G)f Fp(and)h Fo(H)52 b Fp(so)44 b(that)568 4448 y Fo(x)28 b Fm(2)g Fo(G)p Fp(,)35 b Fo(F)41 b Fm(\022)28 b Fo(H)r(:)324 4641 y Fk(Commen)m(t)36 b(5.3)134 b Fp(\(i\))49 b(Every)32 b(metrizable)g(sp)-5 b(ac)g(e)32 b(is)g Fo(T)2427 4656 y Fh(3)2467 4641 y Fp(;)h(for)f(it)h(is)g(c)-5 b(ertainly)32 b Fo(T)3335 4656 y Fh(1)3407 4641 y Fp(and)568 4761 y(given)j Fo(x)29 b Fm(62)65 b Fp(close)-5 b(d)34 b Fo(F)14 b Fp(,)36 b(we)f(have)g Fo(x)30 b Fm(2)65 b Fp(op)-5 b(en)34 b Fo(X)d Fm(n)22 b Fo(F)50 b Fp(so)35 b(ther)-5 b(e)36 b(exists)f Fo(\017)30 b(>)f Fq(0)36 b Fp(so)568 4882 y(that)d Fo(x)28 b Fm(2)g Fo(B)5 b Fq(\()p Fo(x;)17 b(\017)p Fq(\))28 b Fm(\022)h Fo(X)c Fm(n)18 b Fo(F)c Fp(.)44 b(Put)34 b Fo(G)27 b Fq(=)h Fo(B)5 b Fq(\()p Fo(x;)2315 4842 y Fj(\017)p 2312 4858 36 4 v 2312 4916 a Fh(2)2357 4882 y Fq(\))33 b Fp(and)g Fo(H)i Fq(=)27 b Fm(f)p Fo(y)k Fq(:)d Fo(d)p Fq(\()p Fo(x;)17 b(y)t Fq(\))26 b Fo(>)3440 4842 y Fj(\017)p 3437 4858 V 3437 4916 a Fh(2)3482 4882 y Fm(g)p Fp(;)568 5002 y(the)35 b(r)-5 b(esult)35 b(now)f(fol)5 b(lows.)1894 5251 y Fq(46)p eop %%Page: 47 48 47 47 bop 379 548 a Fp(\(ii\))49 b(Obviously)34 b Fo(T)1068 563 y Fh(3)1135 548 y Fm(\))28 b Fo(T)1320 563 y Fh(2)1359 548 y Fp(.)350 749 y(\(iii\))48 b(One)34 b(c)-5 b(an)34 b(devise)g(examples)g(of)g Fo(T)1830 764 y Fh(2)1905 749 y Fp(sp)-5 b(ac)g(es)34 b(which)g(ar)-5 b(e)34 b(not)h Fo(T)2867 764 y Fh(3)2907 749 y Fp(.)364 950 y(\(iv\))49 b(It's)34 b(fairly)h(r)-5 b(outine)35 b(to)g(che)-5 b(ck)34 b(that)h Fo(T)1967 965 y Fh(3)2042 950 y Fp(is)f(pr)-5 b(o)g(ductive)35 b(and)f(her)-5 b(e)g(ditary.)394 1151 y(\(v\))49 b(Warning:)63 b(Some)44 b(b)-5 b(o)g(oks)43 b(take)i Fo(T)1845 1166 y Fh(3)1929 1151 y Fp(to)f(me)-5 b(an)44 b(De\014nition)f(5.3\(ii\))g(alone,)j(and)568 1271 y(r)-5 b(e)g(gular)36 b(to)h(me)-5 b(an)36 b(De\014nition)g (5.3\(i\))g Fq(and)h Fp(\(ii\);)g(others)g(do)f(exactly)h(the)g(opp)-5 b(o-)568 1391 y(site!)324 1723 y Fn(5.4)160 b Fc(T)773 1766 y Fq(3)832 1727 y Fb(1)p 832 1743 42 4 v 832 1801 a(2)942 1723 y Fn(Spaces)324 1982 y Fk(De\014nition)36 b(5.4)49 b Fp(A)25 b(sp)-5 b(ac)g(e)24 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))f Fp(is)f Fo(T)1839 2007 y Fh(3)1884 1980 y Fe(1)p 1885 1992 31 4 v 1885 2033 a(2)1954 1982 y Fp(or)h Fk(completely)e(regular)h Fp(or)h Fk(T)m(yc)m(hono\013)324 2103 y Fp(i\013)409 2299 y(\(i\))49 b(it)35 b(is)f Fo(T)826 2314 y Fh(1)866 2299 y Fp(,)h(and)379 2499 y(\(ii\))49 b(given)38 b Fo(x)f Fm(2)g Fo(X)8 b Fp(,)40 b(close)-5 b(d)38 b(non-empty)h Fo(F)50 b Fm(\022)37 b Fo(X)47 b Fp(such)39 b(that)h Fo(x)d Fm(62)f Fo(F)14 b Fp(,)41 b(ther)-5 b(e)39 b(exists)568 2620 y(c)-5 b(ontinuous)34 b Fo(f)39 b Fq(:)27 b Fo(X)36 b Fm(!)27 b Fq([0)p Fo(;)17 b Fq(1])35 b Fp(such)f(that)h Fo(f)11 b Fq(\()p Fo(F)j Fq(\))27 b(=)h Fm(f)p Fq(0)p Fm(g)34 b Fp(and)h Fo(f)11 b Fq(\()p Fo(x)p Fq(\))27 b(=)h(1)p Fp(.)324 2838 y Fk(Commen)m(t)36 b(5.4)134 b Fp(\(i\))49 b(Every)35 b(metrizable)f(sp)-5 b(ac)g(e)34 b(is)g Fo(T)2436 2863 y Fh(3)2481 2836 y Fe(1)p 2482 2848 V 2482 2889 a(2)379 3053 y Fp(\(ii\))49 b(Every)d Fo(T)915 3078 y Fh(3)960 3050 y Fe(1)p 960 3062 V 960 3104 a(2)1051 3053 y Fp(sp)-5 b(ac)g(e)46 b(is)f Fo(T)1489 3068 y Fh(3)1575 3053 y Fp(\()h(such)g(a)g(sp)-5 b(ac)g(e)46 b(is)f(c)-5 b(ertainly)46 b Fo(T)2839 3068 y Fh(1)2925 3053 y Fp(and)g(given)f Fo(x)k Fm(62)603 3194 y Fp(close)-5 b(d)33 b Fo(F)14 b Fp(,)40 b(cho)-5 b(ose)39 b Fo(f)50 b Fp(as)39 b(in)g(the)g(de\014nition;)h(de\014ne)f Fo(G)d Fq(=)f Fo(f)2908 3158 y Fg(\000)p Fh(1)3002 3194 y Fq(\([0)p Fo(;)3170 3155 y Fh(1)p 3170 3171 36 4 v 3170 3228 a(3)3215 3194 y Fq(\)\))p Fp(,)40 b Fo(H)k Fq(=)568 3314 y Fo(f)627 3278 y Fg(\000)p Fh(1)721 3314 y Fq(\(\()807 3275 y Fh(2)p 807 3291 V 807 3349 a(3)852 3314 y Fo(;)17 b Fq(1]\))34 b Fp(and)g(observe)g(that)i Fo(T)1835 3329 y Fh(3)1909 3314 y Fp(fol)5 b(lows.\))350 3515 y(\(iii\))48 b(Examples)33 b(ar)-5 b(e)35 b(known)f(of)g Fo(T)1643 3530 y Fh(3)1718 3515 y Fp(sp)-5 b(ac)g(es)34 b(which)g(fail)g(to)h(b)-5 b(e)35 b(T)-7 b(ychono\013)364 3716 y(\(iv\))49 b Fo(T)625 3741 y Fh(3)670 3714 y Fe(1)p 670 3726 31 4 v 670 3767 a(2)750 3716 y Fp(is)34 b(pr)-5 b(o)g(ductive)35 b(and)f(her)-5 b(e)g(ditary.)324 4048 y Fn(5.5)160 b Fc(T)773 4069 y Fq(4)880 4048 y Fn(Spaces)324 4267 y Fk(De\014nition)36 b(5.5)49 b Fp(A)35 b(sp)-5 b(ac)g(e)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))35 b Fp(is)f Fo(T)1879 4282 y Fh(4)1954 4267 y Fp(or)g Fk(normal)g Fp(if)409 4463 y(\(i\))49 b(it)35 b(is)f Fo(T)826 4478 y Fh(1)866 4463 y Fp(,)h(and)379 4663 y(\(ii\))49 b(given)32 b(disjoint)h(non-empty)g(close)-5 b(d)33 b(subsets)h Fo(A)p Fp(,)g Fo(B)k Fp(of)c Fo(X)8 b Fp(,)33 b(ther)-5 b(e)34 b(exist)f(disjoint)568 4784 y(op)-5 b(en)34 b(sets)h Fo(G)p Fp(,)f Fo(H)43 b Fp(such)34 b(that)i Fo(A)27 b Fm(\022)i Fo(G)p Fp(,)34 b Fo(B)f Fm(\022)28 b Fo(H)8 b Fp(.)324 5002 y Fk(Theorem)37 b(5.7)49 b Fp(Every)35 b(metrizable)f(sp)-5 b(ac)g(e)34 b Fq(\()p Fo(X)r(;)17 b Fm(T)26 b Fq(\))34 b Fp(is)h Fo(T)2470 5017 y Fh(4)2510 5002 y Fp(.)1894 5251 y Fq(47)p eop %%Page: 48 49 48 48 bop 324 548 a Fq(Pro)s(of)p 324 561 235 4 v 39 w(Certainly)-8 b(,)41 b Fo(X)47 b Fq(is)40 b Fo(T)1348 563 y Fh(1)1387 548 y Fq(;)k(c)m(ho)s(ose)d(a)e(metric)g Fo(d)h Fq(on)f Fo(X)48 b Fq(suc)m(h)41 b(that)f Fm(T)65 b Fq(is)40 b Fm(T)3261 563 y Fj(d)3301 548 y Fq(.)66 b(The)324 668 y(distance)33 b(of)f(a)g(p)s(oin)m(t)g Fo(p)g Fq(from)g(a)g(non-empt)m(y)h(set)g Fo(A)g Fq(can)f(b)s(e)h (de\014ned)h(th)m(us:)1323 883 y Fo(d)p Fq(\()p Fo(p;)17 b(A)p Fq(\))27 b(=)g(inf)6 b Fm(f)p Fo(d)p Fq(\()p Fo(p;)17 b(a)p Fq(\))27 b(:)h Fo(a)g Fm(2)g Fo(A)p Fm(g)324 1098 y Fq(Giv)m(en)k(disjoin)m(t)g(non-empt)m(y)g(closed)h(sets)h Fo(A)p Fq(,)f Fo(B)5 b Fq(,)32 b(let)568 1297 y Fo(G)27 b Fq(=)h Fm(f)p Fo(x)g Fq(:)g Fo(d)p Fq(\()p Fo(x;)17 b(A)p Fq(\))27 b Fo(<)h(d)p Fq(\()p Fo(x;)17 b(B)5 b Fq(\))p Fm(g)568 1499 y Fo(H)35 b Fq(=)27 b Fm(f)p Fo(x)h Fq(:)g Fo(d)p Fq(\()p Fo(x;)17 b(B)5 b Fq(\))28 b Fo(<)f(d)p Fq(\()p Fo(x;)17 b(A)p Fq(\))p Fm(g)p Fq(.)324 1697 y(Clearly)-8 b(,)25 b Fo(G)6 b Fm(\\)g Fo(H)35 b Fq(=)28 b Fm(;)p Fq(.)40 b(Also,)26 b(eac)m(h)f(is)f(op)s(en)h(\(if)e Fo(x)28 b Fm(2)g Fo(G)d Fq(and)g Fo(\017)j Fq(=)2697 1658 y Fh(1)p 2697 1674 36 4 v 2697 1732 a(2)2742 1697 y Fm(f)p Fo(d)p Fq(\()p Fo(x;)17 b(B)5 b Fq(\))h Fm(\000)g Fo(d)p Fq(\()p Fo(x;)17 b(A)p Fq(\))p Fm(g)p Fq(,)324 1818 y(then)36 b Fo(B)5 b Fq(\()p Fo(x;)17 b(\017)p Fq(\))34 b Fm(\022)f Fo(G)p Fq(,)k(b)m(y)f(the)h(triangle)c(inequalit)m(y)-8 b(.\))52 b(No)m(w,)37 b(if)e Fo(d)p Fq(\()p Fo(p;)17 b(A)p Fq(\))32 b(=)h(0,)k(then)f(for)324 1938 y(all)k Fo(n)45 b Fm(2)g Fo(N)10 b Fq(,)46 b(there)d(exists)g Fo(x)1437 1953 y Fj(n)1529 1938 y Fm(2)i Fo(A)e Fq(suc)m(h)h(that)e Fo(d)p Fq(\()p Fo(p;)17 b(x)2444 1953 y Fj(n)2491 1938 y Fq(\))44 b Fo(<)2708 1899 y Fh(1)p 2704 1915 43 4 v 2704 1973 a Fj(n)2757 1938 y Fq(.)73 b(So)42 b Fo(d)p Fq(\()p Fo(p;)17 b(x)3239 1953 y Fj(n)3286 1938 y Fq(\))45 b Fm(!)f Fq(0)324 2059 y(i.e.)f Fo(x)547 2074 y Fj(n)642 2059 y Fm(!)j Fo(p)p Fq(,)h(whence)f Fo(p)h Fm(2)1499 2033 y Fq(\026)1473 2059 y Fo(A)q Fq(.)77 b(Th)m(us)46 b(for)d(eac)m(h)i Fo(x)i Fm(2)h Fo(A)p Fq(,)f Fo(x)g Fm(62)h Fo(B)k Fq(=)3152 2033 y(\026)3129 2059 y Fo(B)d Fq(so)44 b(that)324 2179 y Fo(d)p Fq(\()p Fo(x;)17 b(B)5 b Fq(\))27 b Fo(>)h Fq(0)f(=)h Fo(d)p Fq(\()p Fo(x;)17 b(A)p Fq(\))32 b(i.e.)g Fo(x)c Fm(2)h Fo(G)p Fq(.)43 b(Hence)34 b Fo(A)28 b Fm(\022)g Fo(G)p Fq(.)43 b(Similarly)29 b Fo(B)k Fm(\022)28 b Fo(H)8 b Fq(.)324 2299 y(It's)36 b(true)f(that)g Fo(T)977 2314 y Fh(4)1049 2299 y Fm(\))d Fo(T)1238 2324 y Fh(3)1283 2297 y Fe(1)p 1283 2309 31 4 v 1283 2350 a(2)1363 2299 y Fq(but)j(not)g(v)m(ery)i(ob)m(vious.)51 b(First)34 b(note)i(that)f(if)f Fo(G)3178 2314 y Fh(0)3217 2299 y Fq(,)i Fo(G)3357 2314 y Fh(1)3432 2299 y Fq(are)324 2440 y(op)s(en)e(in)f(a)h Fo(T)816 2455 y Fh(4)889 2440 y Fq(space)h(with)1408 2415 y(\026)1375 2440 y Fo(G)1452 2455 y Fh(0)1521 2440 y Fm(\022)30 b Fo(G)1705 2455 y Fh(1)1745 2440 y Fq(,)k(then)g(there)h(exists)g(op)s(en)f Fo(G)2875 2438 y Fe(1)p 2875 2450 V 2875 2491 a(2)2953 2440 y Fq(with)3210 2415 y(\026)3176 2440 y Fo(G)3253 2455 y Fh(0)3323 2440 y Fm(\022)c Fo(G)3517 2438 y Fe(1)p 3517 2450 V 3517 2491 a(2)324 2581 y Fq(and)559 2556 y(\026)517 2581 y Fo(G)604 2579 y Fe(1)p 604 2591 V 604 2632 a(2)683 2581 y Fm(\022)35 b Fo(G)872 2596 y Fh(1)948 2581 y Fq(\(b)s(ecause)j(the)f(giv)m(en)1816 2556 y(\026)1782 2581 y Fo(G)1859 2596 y Fh(0)1935 2581 y Fq(and)g Fo(X)32 b Fm(n)25 b Fo(G)2394 2596 y Fh(1)2470 2581 y Fq(are)36 b(disjoin)m(t)g(closed)h(sets)g(so)324 2722 y(that)f(there)g(exist)h (disjoin)m(t)e(op)s(en)h(sets)h Fo(G)1897 2720 y Fe(1)p 1897 2732 V 1897 2773 a(2)1941 2722 y Fq(,)g Fo(H)44 b Fq(suc)m(h)37 b(that)2602 2696 y(\026)2568 2722 y Fo(G)2645 2737 y Fh(0)2718 2722 y Fm(\022)d Fo(G)2916 2720 y Fe(1)p 2916 2732 V 2916 2773 a(2)2961 2722 y Fq(,)j Fo(X)32 b Fm(n)24 b Fo(G)3289 2737 y Fh(1)3362 2722 y Fm(\022)34 b Fo(H)324 2856 y Fq(i.e.)e Fo(G)558 2871 y Fh(1)625 2856 y Fm(\023)61 b Fq(\(closed\))32 b Fo(X)e Fm(n)22 b Fo(H)35 b Fm(\023)29 b Fo(G)1618 2854 y Fe(1)p 1617 2866 V 1617 2907 a(2)1662 2856 y Fq(\).)324 3092 y Fk(Lemma)37 b(5.1)g(\()p Fp(Urysohn)-10 b('s)35 b(L)-5 b(emma\))48 b(L)-5 b(et)50 b Fo(F)2040 3107 y Fh(1)2079 3092 y Fp(,)j Fo(F)2225 3107 y Fh(2)2314 3092 y Fp(b)-5 b(e)50 b(disjoint)f (non-empty)g(close)-5 b(d)324 3212 y(subsets)36 b(of)g(a)g Fo(T)913 3227 y Fh(4)989 3212 y Fp(sp)-5 b(ac)g(e;)37 b(then)f(ther)-5 b(e)36 b(exists)g(a)h(c)-5 b(ontinuous)36 b(function)g Fo(f)41 b Fq(:)31 b Fo(X)38 b Fm(!)30 b Fq([0)p Fo(;)17 b Fq(1])324 3332 y Fp(such)34 b(that)i Fo(f)11 b Fq(\()p Fo(F)905 3347 y Fh(1)944 3332 y Fq(\))28 b(=)f Fm(f)p Fq(0)p Fm(g)p Fp(,)34 b Fo(f)11 b Fq(\()p Fo(F)1486 3347 y Fh(2)1526 3332 y Fq(\))27 b(=)h Fm(f)p Fq(1)p Fm(g)p Fp(.)324 3555 y Fq(Pro)s(of)p 324 3568 235 4 v 23 w(Giv)m(en)c(disjoin)m(t)f(closed)i Fo(F)1538 3570 y Fh(1)1601 3555 y Fq(and)g Fo(F)1846 3570 y Fh(2)1885 3555 y Fq(,)h(c)m(ho)s(ose)f(disjoin)m(t)e(op)s(en)h Fo(G)2882 3570 y Fh(0)2946 3555 y Fq(and)g Fo(H)3208 3570 y Fh(0)3271 3555 y Fq(so)h(that)324 3675 y Fo(F)387 3690 y Fh(1)454 3675 y Fm(\022)j Fo(G)636 3690 y Fh(0)676 3675 y Fq(,)c Fo(F)790 3690 y Fh(2)858 3675 y Fm(\022)k Fo(H)1044 3690 y Fh(0)1083 3675 y Fq(.)40 b(De\014ne)23 b Fo(G)1519 3690 y Fh(1)1586 3675 y Fq(=)28 b Fo(X)10 b Fm(n)r Fo(F)1896 3690 y Fh(2)1957 3675 y Fq(\(op)s(en\).)41 b(Since)22 b Fo(G)2625 3690 y Fh(0)2693 3675 y Fm(\022)60 b Fq(\(closed\))33 b Fo(X)10 b Fm(n)r Fo(H)3418 3690 y Fh(0)3484 3675 y Fm(\022)324 3795 y Fo(X)30 b Fm(n)22 b Fo(F)570 3810 y Fh(2)637 3795 y Fq(=)27 b Fo(G)817 3810 y Fh(1)857 3795 y Fq(,)32 b(w)m(e)i(ha)m(v)m(e)1319 3770 y(\026)1285 3795 y Fo(G)1362 3810 y Fh(0)1429 3795 y Fm(\022)28 b Fo(G)1611 3810 y Fh(1)1651 3795 y Fq(.)324 3916 y(By)33 b(the)g(previous)g(remark,)f(w)m(e)i(can)f(no)m(w)g (construct:)416 4115 y(\(i\))48 b Fo(G)655 4113 y Fe(1)p 655 4125 31 4 v 655 4166 a(2)727 4115 y Fm(2)28 b(T)d Fq(:)1005 4089 y(\026)971 4115 y Fo(G)1048 4130 y Fh(0)1115 4115 y Fm(\022)j Fo(G)1307 4113 y Fe(1)p 1307 4125 V 1307 4166 a(2)1352 4115 y Fq(,)1453 4089 y(\026)1411 4115 y Fo(G)1498 4113 y Fe(1)p 1498 4125 V 1498 4166 a(2)1570 4115 y Fm(\022)h Fo(G)1753 4130 y Fh(1)1792 4115 y Fq(.)389 4337 y(\(ii\))47 b Fo(G)655 4335 y Fe(1)p 655 4347 V 655 4388 a(4)699 4337 y Fq(,)33 b Fo(G)846 4335 y Fe(3)p 846 4347 V 846 4388 a(4)918 4337 y Fm(2)28 b(T)e Fq(:)1196 4312 y(\026)1162 4337 y Fo(G)1239 4352 y Fh(0)1306 4337 y Fm(\022)i Fo(G)1498 4335 y Fe(1)p 1498 4347 V 1498 4388 a(4)1543 4337 y Fq(,)1644 4312 y(\026)1602 4337 y Fo(G)1689 4335 y Fe(1)p 1689 4347 V 1689 4388 a(4)1761 4337 y Fm(\022)h Fo(G)1954 4335 y Fe(1)p 1954 4347 V 1954 4388 a(2)1998 4337 y Fq(,)2099 4312 y(\026)2058 4337 y Fo(G)2145 4335 y Fe(1)p 2145 4347 V 2145 4388 a(2)2217 4337 y Fm(\022)f Fo(G)2409 4335 y Fe(3)p 2409 4347 V 2409 4388 a(4)2453 4337 y Fq(,)2554 4312 y(\026)2513 4337 y Fo(G)2600 4335 y Fe(3)p 2600 4347 V 2600 4388 a(4)2672 4337 y Fm(\022)g Fo(G)2854 4352 y Fh(1)2894 4337 y Fq(.)362 4552 y(\(iii\))46 b(.)16 b(.)g(.)g(and)33 b(so)g(on!)324 4751 y(Th)m(us)h(w)m(e)g(get)e(an)h (indexed)g(family)d(of)i(op)s(en)h(sets)1236 4984 y Fm(f)p Fo(G)1363 4999 y Fj(r)1428 4984 y Fq(:)28 b Fo(r)j Fq(=)1676 4916 y Fo(m)p 1671 4960 96 4 v 1671 5052 a Fq(2)1720 5023 y Fj(n)1777 4984 y Fo(;)17 b Fq(0)27 b Fm(\024)h Fo(m)g Fm(\024)g Fq(2)2269 4942 y Fj(n)2316 4984 y Fo(;)17 b(n)28 b Fm(\025)g Fq(1)p Fm(g)1894 5251 y Fq(48)p eop %%Page: 49 50 49 49 bop 324 548 a Fq(suc)m(h)34 b(that)e Fo(r)799 563 y Fh(1)866 548 y Fm(\024)c Fo(r)1015 563 y Fh(2)1082 548 y Fm(\))1259 523 y Fq(\026)1210 548 y Fo(G)1287 563 y Fj(r)1319 572 y Fe(1)1385 548 y Fm(\022)g Fo(G)1567 563 y Fj(r)1599 572 y Fe(2)1638 548 y Fq(.)324 668 y(Observ)m(e)42 b(that)d(the)h(index)h(set)f(is)f(dense)j(in)d([0)p Fo(;)17 b Fq(1]:)57 b(if)39 b Fo(s)h(<)g(t)g Fq(in)f([0)p Fo(;)17 b Fq(1],)41 b(there)g(exists)324 789 y(some)586 749 y Fj(m)p 578 765 78 4 v 578 823 a Fh(2)613 804 y Fi(n)698 789 y Fq(suc)m(h)34 b(that)f Fo(s)27 b(<)1325 749 y Fj(m)p 1317 765 V 1317 823 a Fh(2)1352 804 y Fi(n)1432 789 y Fo(<)h(t)p Fq(.)44 b(De\014ne)1188 1080 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))27 b(=)1509 934 y Ff(\()1617 1019 y Fq(inf)6 b Fm(f)p Fo(r)31 b Fq(:)c Fo(x)h Fm(2)h Fo(G)2169 1034 y Fj(r)2206 1019 y Fm(g)83 b Fo(x)28 b Fm(62)h Fo(F)2580 1034 y Fh(2)1617 1140 y Fq(1)673 b Fo(x)28 b Fm(2)h Fo(F)2580 1155 y Fh(2)2619 1140 y Fo(:)324 1363 y Fq(Certainly)34 b Fo(f)41 b Fq(:)31 b Fo(X)39 b Fm(!)30 b Fq([0)p Fo(;)17 b Fq(1],)35 b Fo(f)11 b Fq(\()p Fo(F)1572 1378 y Fh(2)1611 1363 y Fq(\))31 b(=)g Fm(f)p Fq(1)p Fm(g)p Fq(,)j Fo(f)11 b Fq(\()p Fo(F)2157 1378 y Fh(1)2196 1363 y Fq(\))31 b(=)g Fm(f)p Fq(0)p Fm(g)p Fq(.)49 b(T)-8 b(o)34 b(sho)m(w)i Fo(f)45 b Fq(con)m(tin)m(uous,)324 1483 y(it)31 b(su\016ces)k(to)d(sho) m(w)i(that)e Fo(f)1384 1447 y Fg(\000)p Fh(1)1478 1483 y Fq(\([0)p Fo(;)17 b(\013)q Fq(\)\))32 b(and)h Fo(f)2056 1447 y Fg(\000)p Fh(1)2150 1483 y Fq(\(\()p Fo(\013)q(;)17 b Fq(1]\))31 b(are)i(op)s(en)g(for)f(0)27 b Fo(<)h(\013)g(<)f Fq(1.)324 1603 y(W)-8 b(ell,)43 b Fo(f)11 b Fq(\()p Fo(x)p Fq(\))44 b Fo(<)g(\013)f Fq(i\013)e(there)i(exists)g(some)f Fo(r)k Fq(=)2190 1564 y Fj(m)p 2182 1580 V 2182 1638 a Fh(2)2217 1619 y Fi(n)2312 1603 y Fq(suc)m(h)d(that)f Fo(f)11 b Fq(\()p Fo(x)p Fq(\))44 b Fo(<)g(r)j(<)d(\013)q Fq(.)72 b(It)324 1724 y(follo)m(ws)31 b(that)h Fo(f)914 1688 y Fg(\000)p Fh(1)1008 1724 y Fq(\([0)p Fo(;)17 b(\013)q Fq(\)\))27 b(=)h Fm([)1502 1739 y Fj(r)r(<\013)1640 1724 y Fo(G)1717 1739 y Fj(r)1755 1724 y Fq(,)33 b(a)f(union)g(of)g(op)s(en) h(sets.)324 1844 y(Again,)d Fo(f)11 b Fq(\()p Fo(x)p Fq(\))27 b Fo(>)h(\013)j Fq(i\013)e(there)i(exist)g Fo(r)1682 1859 y Fh(1)1721 1844 y Fq(,)g Fo(r)1823 1859 y Fh(2)1893 1844 y Fq(suc)m(h)g(that)f Fo(\013)f(<)e(r)2557 1859 y Fh(1)2624 1844 y Fo(<)h(r)2772 1859 y Fh(2)2839 1844 y Fo(<)g(f)11 b Fq(\()p Fo(x)p Fq(\),)30 b(implying)324 1965 y(that)k Fo(x)e Fm(62)g Fo(G)799 1980 y Fj(r)831 1989 y Fe(2)904 1965 y Fq(whence)37 b Fo(x)31 b Fm(62)1482 1939 y Fq(\026)1432 1965 y Fo(G)1509 1980 y Fj(r)1541 1989 y Fe(1)1580 1965 y Fq(.)50 b(It)34 b(follo)m(ws)g(that)g Fo(f)2359 1928 y Fg(\000)p Fh(1)2453 1965 y Fq(\(\()p Fo(\013)q(;)17 b Fq(1]\))31 b(=)g Fm([)2954 1980 y Fj(r)2986 1989 y Fe(1)3021 1980 y Fj(>\013)3125 1965 y Fq(\()p Fo(X)h Fm(n)3398 1939 y Fq(\026)3349 1965 y Fo(G)3426 1980 y Fj(r)3458 1989 y Fe(1)3497 1965 y Fq(\),)324 2085 y(whic)m(h)h(is)f(again)f(op)s(en.)324 2313 y Fk(Corollary)36 b(5.4)49 b Fp(Every)35 b Fo(T)1350 2328 y Fh(4)1424 2313 y Fp(sp)-5 b(ac)g(e)34 b(is)h Fo(T)1840 2338 y Fh(3)1885 2311 y Fe(1)p 1885 2323 31 4 v 1885 2364 a(2)1930 2313 y Fp(.)324 2555 y Fq(Pro)s(of)p 324 2568 235 4 v 35 w(Immediate)f(from) g(Lemma)g(5.1.)52 b(\(Note)36 b(that)f(there)h(exist)g(spaces)h(whic)m (h)g(are)324 2676 y Fo(T)381 2701 y Fh(3)426 2673 y Fe(1)p 426 2685 31 4 v 426 2727 a(2)503 2676 y Fq(but)c(not)g Fo(T)913 2691 y Fh(4)952 2676 y Fq(.\))324 2879 y Fk(Theorem)k(5.8)49 b Fp(A)n(ny)35 b(c)-5 b(omp)g(act)34 b Fo(T)1631 2894 y Fh(2)1705 2879 y Fp(sp)-5 b(ac)g(e)34 b(is)h Fo(T)2121 2894 y Fh(4)2161 2879 y Fp(.)324 3082 y Fq(Pro)s(of)p 324 3095 235 4 v 32 w(Use)e(Corollary)e(5.2)h(to)g(Theorem)h(5.3.)324 3203 y Fp(Note)43 b Fq(Unlik)m(e)g(the)h(previous)f(axioms,)i Fo(T)1886 3218 y Fh(4)1969 3203 y Fq(is)d(neither)i(hereditary)f(nor)g (pro)s(ductiv)m(e.)324 3323 y(The)g(global)d(view)j(of)f(the)h(hierarc) m(h)m(y)h(can)f(no)m(w)g(b)s(e)g(\014lled)e(in)h(as)g(an)h(exercise)h (from)324 3444 y(data)32 b(supplied)g(ab)s(o)m(v)m(e:-)374 3556 y Fa(Metrizable)98 b Fp(Her)-5 b(e)g(ditary?)100 b(Pr)-5 b(o)g(ductive?)374 3676 y Fo(T)431 3691 y Fh(4)374 3796 y Fo(T)431 3821 y Fh(3)476 3794 y Fe(1)p 476 3806 31 4 v 476 3847 a(2)374 3932 y Fo(T)431 3947 y Fh(3)374 4052 y Fo(T)431 4067 y Fh(2)374 4172 y Fo(T)431 4187 y Fh(1)324 4286 y Fq(The)43 b(follo)m(wing)c(is)j(presen)m(ted)j(as)d (an)g(indication)e(of)i(ho)m(w)g(close)h(w)m(e)g(are)f(to)g(ha)m(ving) 324 4406 y(`come)32 b(full)f(circle'.)324 4610 y Fk(Theorem)37 b(5.9)49 b Fp(A)n(ny)35 b(c)-5 b(ompletely)34 b(sep)-5 b(ar)g(able)34 b Fo(T)2143 4625 y Fh(4)2217 4610 y Fp(sp)-5 b(ac)g(e)34 b(is)h(metrizable!)324 4813 y Fq(Sk)m(etc)m(h)f(Pro)s(of)p 324 4826 546 4 v 324 4933 a(Cho)s(ose)i(a)f(coun)m(table)h(base;)i (list)c(as)h Fm(f)p Fq(\()p Fo(G)1897 4948 y Fj(n)1944 4933 y Fo(;)17 b(H)2069 4948 y Fj(n)2116 4933 y Fq(\))32 b(:)h Fo(n)g Fm(\025)g Fq(1)p Fm(g)i Fq(those)h(pairs)f(of)g(elemen)m (ts)1894 5251 y(49)p eop %%Page: 50 51 50 50 bop 324 548 a Fq(of)27 b(the)h(base)g(for)f(whic)m(h)1261 523 y(\026)1223 548 y Fo(G)1300 563 y Fj(n)1375 548 y Fm(\022)h Fo(H)1561 563 y Fj(n)1608 548 y Fq(.)42 b(F)-8 b(or)27 b(eac)m(h)h Fo(n)p Fq(,)h(use)g(Lemma)d(5.1)h(to)g(get)h(con)m (tin)m(uous)324 668 y Fo(f)372 683 y Fj(n)447 668 y Fq(:)f Fo(X)36 b Fm(!)27 b Fq([0)p Fo(;)17 b Fq(1])32 b(suc)m(h)i(that)e Fo(f)1452 683 y Fj(n)1499 668 y Fq(\()1575 643 y(\026)1537 668 y Fo(G)1614 683 y Fj(n)1661 668 y Fq(\))c(=)f Fm(f)p Fq(0)p Fm(g)p Fq(,)32 b Fo(f)2086 683 y Fj(n)2134 668 y Fq(\()p Fo(X)d Fm(n)22 b Fo(H)2435 683 y Fj(n)2482 668 y Fq(\))28 b(=)f Fm(f)p Fq(1)p Fm(g)p Fq(.)43 b(De\014ne)1250 974 y Fo(d)p Fq(\()p Fo(x;)17 b(y)t Fq(\))27 b(=)1658 808 y Ff(v)1658 854 y(u)1658 904 y(u)1658 954 y(t)p 1746 808 863 4 v 1753 891 a(X)1746 1074 y Fj(n)p Fg(\025)p Fh(1)1879 974 y Fm(f)1939 907 y Fo(f)1987 922 y Fj(n)2034 907 y Fq(\()p Fo(x)p Fq(\))22 b Fm(\000)h Fo(f)2335 922 y Fj(n)2382 907 y Fq(\()p Fo(y)t Fq(\))p 1939 951 571 4 v 2176 1042 a(2)2225 1014 y Fj(n)2519 974 y Fm(g)2569 945 y Fh(2)2608 974 y Fo(:)324 1280 y Fq(One)33 b(con\014rms)f(that)h Fo(d)f Fp(is)g Fq(a)h(metric,)e Fp(and)h Fq(induces)h(the)g(original)d (top)s(ology)-8 b(.)1894 5251 y(50)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF