Volume 9, 1984 Pges 149 163 http://topology.uburn.edu/tp/ SPACES DOMINATED BY METRIC SUBSETS by Yoshio Tnk nd Zhou Ho-xun Topology Proceedings Web: http://topology.uburn.edu/tp/ Mil: Topology Proceedings Deprtment of Mthemtics & Sttistics Auburn University, Albm 36849, USA E-mil: topolog@uburn.edu ISSN: 0146-4124 COPYRIGHT c by Topology Proceedings. All rights reserved.
TOPOLOGY PROCEEDINGS Volume 9 1984 149 SPACES DOMINATED BY METRIC SUBSETS Yoshio Tnk nd Zhou Ho-xun Introduction Let X be spce nd J be closed cover of X. Then X is determined by Jl, if A c X is closed in X whenever A n F is reltively closed in F for ech F E J. A spce X is k-spce (resp. sequentiz spce) if it is determined by the cover of ll compct subsets (resp. compct metric subsets). Recll tht X is dominted by J 2, if the union of ny subcollection J' of J is closed in X nd the union is determined by J'. We remrk tht if X is dominted by J, then it is determined by J, but the converse does not hold. In cse tht the closed cover J is incresing nd countble, the converse holds. Every CW-complex, more generlly every chunk-complex [2] is dominted by cover of compct metric subsets. Now, let X be regulr spce determined by pointcountble closed cover of seprble metric subsets. In [14] nd [15], the first uthor showed respectively tht X is Frechet if nd only if it is Lsnev, nd tht X is metric if nd only if it contins no closed copy of sequentil fn 8 nd no Arens' spce 8 2. Recll tht spce w X is Frechet, if for every A c X nd every x E A there lfollowing [7], we use "X is determined by J" insted of the usul "X hs the wek topology with respect to (or determined by) J. 2rn some literture, "X hs the wek (hereditrily wek; or Whitehed wek) topology with respect to J" is used insted of "X is dominted by J."
150 Tnk nd Ho-xun exists sequence in A converging to the point x. Also, spce X is Lsnev if X is the closed imge of metric spce. The spce Sw is the quotient spce obtined from the topologicl sum of countbly mny convergent sequences by identifying ll the limit' points. As for the spce S?, e.g., see [4; Exmple 1.6.19]. In this pper, we shll give some nlogous results spces re dominted by metric subsets, nd investigte --chunk-complexes nd CW-complexes s spces dominted by compct metric subsets. In Section 1, we show tht every Frechet spce dominted by metric subsets is Lsnev, nd tht every spce dominted by metric subsets is metric if nd only if it contins no closed copy of Sw nd no S2. The former is n ffirmtive nswer to question in [14; Problem 2.4], nd the ltter ws proved in [5] for every spce dominted by countbly mny compct metric subsets. In Section 2, we show tht every Frechet spce dominted by compct metric subsets is equivlent to Frechet chunkcomplex, nd such complex is chrcterized s the closed 3 imge of loclly compct metric spce. In Section 3, we show tht every Frechet CW-complex is especilly the closed "'c.' imge of the topologicl sum of Eucliden simplexes (hence, the closed imge of loclly compct metric CW-complex). We ssume tht ll spces re Husdorff nd ll mps re continuous nd onto. 3After prepring this pper, M. Ito independently proved tht every Frechet chunk-complex is Lsnev.
TOPOLOGY PROCEEDINGS Volume 9 1984 151 1. Spces Dominted by Metric Subsets nd L;nev Spces Lemm 1.1. Suppose tht X is Fp~chet spce dominted by closed covep {X; < S}. Let Y = X - Uo<Xo' nd Y~ Y x {}. Let X* be the topologicl sum LY~. Then the obvious mp f: X* ~ X is closed. Ppoof Since f is clerly continuous, to prove tht f is closed, it is sufficient to show tht f(f) c f (F) for ny F c X*. Let x E f(f) - f (F). Since X is Frechet, there exist distinct points x. E f{f) with x. ~ x. Choose 1 1-1 point Pi E f (xi) n F for ech i E N, nd let P = {p.; i EN}. 1 Suppose tht P is not contined in ny finite union of Y~'s. Let = min{; Y~ n P ~ ~} nd l choose point p. 1 1 By induction, we cn choose countbly mny ordinls n = rnin{; Y~ n {Pi; i > in-i} ~ ~, ~ l,---, - l } nd countbly mny points Pi E y* with n n n in < i + Then for ech n E N, n < + l nd the subset n l n Y of X contins x. = f (p. ). Since X is Frechet, for 1 1 n n n ech n E N there exist Ynk E X :n Let A = {Y nk ; n,k E N} - {x}. Since x E A, there exists sequence B in A converging to the point x. Let C = U{X n n E N}. Then B c C nd B n X is finite for ech n E N. n But the closed subset C of X is determined by {X ; n EN}. n Then B is closed in C, hence in X. Thus, x E B, contrdiction. Hence the subset P of X* is contined in finite union of Y*'s. So, infinitely mny points Pn. re in some J y*. But fly* is homeomorphism nd x f(p ) E Y n. n. 0 0 J J 0
152 Tnk nd Ho-xun Thus, since x. ~ x, y for some y y* with f(y) = x. n Pn. ~ J J 0 Then, since Pn. E f-l(x ) n F, y E P, so tht x E f(f). J n. J Thus, f(f) c f (P) Tht completes the proof. By Lemm 1.1, we hve the following ffirmtive nswer to the question whether every Frechet spce dominted by compct metric subsets (e.g., Frechet CW-complex) is Lsnev; see [14; Problem 2.4]. Theorem 1.2. Every Frechet spce dominted by closed cover of metric (resp. loclly compct metric) subsets is Lsnev spce (resp. the closed imge of loclly compct metric spce). Theorem 1.3. Lsnev subsets. Frechet. Let X be dominted by closed cover of Then X is L~nev if nd only if it is Proof. The "only if" prt follows from the esy fct tht every Lsnev spce is Frechet. The subsets X of X in Lemm 1.1 re Lsnev, so re the subsets y~ of X*. Thus the "if" prt is routinely verified by Lemm 1.1. Not every countble CW-complex is decomposed into metric subset nd -discrete subset; see [14; p. 284]. But, mong Frechet spces, we hve the following decomposition theorem by Theorem 1.2 nd [8; Theorem 2] (resp. [11; Theorem 4]). Theorem 1.4.* Every Frechet spce dominted by closed cover of metric (res,p. loclly compct metric) subsets is * See note dded in proof.
TOPOLOGY PROCEEDINGS Volume 9 1984 153 decomposed into metric (resp. loclly compct metric) subset nd -discrete (resp. closed discrete) subset. In [5], the metrizbility of spce dominted by countbly mny compct metric subsets is chrcterized by whether or not it contins two copies of the spces Sw nd S2. As for the metrizbility of spce dominted by metric subsets, nlogously we hve Theorem 1.5. Let X be dominted by c~osed cover of metric (resp. loclly compct metric) subsets. Then X is metric (resp. loclly compct metric) if nd only if it contins no closed copy of Sw nd no S2. Proof. The "only if" prt is obvious, so we will prove the "if" prt. Since X is dominted by metric subsets, by [10; Theorem 3] X is regulr spce in which every point is Go. Now, X is sequentil spce which contins no closed copy of Sw nd no S2. Thus, by [15; Theorem 3.1], X is strongly Frecheti tht is, if whenever x ~ n with A + l C An' then there exists x An with x ~ x. Since n n n X is Frechet, by Theorem 1.2, X is the closed imge of metric (resp. loclly compct metric) spce. Then, since X is strongly Frechet, X is metric (resp. loclly compct metric) by [9; Corollry 9.10]. In concluding this section, let us consider the products of spces dominted by metric subsets. For c = 2 w, the quotient spce Sc is similrly defined s the spce Sw.
154 Tnk nd Ho-xun Theorem 1.6. metric subsets. Let X be dominted by closed cover of Then the following re equivlent. subsets. (1) X is loclly compct metric spce. (2) X x S is k-spce. c (3) X x S is dominted by closed cover of metric c Proof (3) + (2) is esily proved. (1) + (3). Let J be loclly finite cover of compct metric subsets of X, nd C be the obvious cover of the infinite convergent sequences in Sc. Since S is dominted c by C, from [10; Theorem 1], X x S is dominted by cover c J x Cof compct metric subsets. (2) + (1). Suppose tht X contins closed copy P of Sw or S2. Note tht the spce S2 is the perfect preimge of Sw. Thus, P x Sc is the perfect pre-imge of Sw x Sc. Since P x Sc is k-spce, so is Sw x Sc. But, the exmple [3; p. 563] implictes tht Sw x Sc is not k-spce. This contrdiction implies tht X contins no closed copy of 5 nd no 52. Then X is metric by Theorem w 1.5. Thus ech point of X x Sc is Go. Then k-spce X x Sc is sequentil by [9; Theorem 7.3]. Thus metric spce X is loclly compct by [12; Theorem 1.1]. Theorem 1.7. Let X be dominted by closed cover of metric subsets. Then (1) nd (2) below hold. 2 (1) Let X be Frechet spce. Then x is k-spce if nd only if X is metric, or X is dominted by countble closed cover of loclly compct metric subsets. (2) X W is k-spce if nd only if X is metric.
TOPOLOGY PROCEEDINGS Volume 9 1984 155 Proof (1) Since X is Lsnev by Theorem 1.2, (1) follows from [6; Theorem 2.15]. (2) Let X W be k-spce. (5)w is not k-spce by W [12; Proposition 4.2]. While, (S2)w is the perfect pre-imge of (5 )W. Then X contins no closed copy of Sw nd no S2. w Hence X is metric by Theorem 1.5. 2. Chunk-Complexes nd Closed Imges ofloclly Compct Metric Spces Recll tht chunk-complex [2} is spce determined by cover H of compct metric subsets, clled chunks, such tht for 5, T E H, either 5 n T = ~ or 5 n T E H, nd for 5 E H, {T E H; T c 5} is finite. A chunk-complex is n Ml-spce [2] dominted by its chunks. Every CW-complex nd every loclly compct metric spce is chunk-complex. Let us cll chunk-complex loclly countble, if it hs loclly countble cover of chunks. Lemm 2.1. Every spce determined by loclly countble cover J of compct metric subsets is loclly countble chunk-complex. Proof. For ny A J, {F E J; A n F ~ ~} is countble. Then, since X is determined by J, by the proof of () ~ (c) of [13; Theorem 1], X is the topologicl sum of spces x( B) determined by countbly mny compct metric subsets XSn(n E N). We cn ssume tht X Sn c for S E B X Sn + l nd n E N. Then X is ~oclly countble chunk-complex with chunks X. sn
156 Tnk nd Ho-xun The regulr, seprble, non-lindelof spce Y of [7; Exmple 9.3] shows tht every spce determined by pointfinite cover of compct metric subsets is not Lsnev, nor chunk-complex. But, mong Frechet spces, we hve Theorem 2.2. Let X be regulr Frechet spce. Then the following re equivlent. (1) X is determined by point-countble cover of compct metric subsets. (2) X is loclly countble chunk-complex. (3) X is the quotient s-imge of loclly compct metric spce. (4) X is the closed s-imge of loclly compct metric spce. Proof. The equivlence (1) ++ (3) ++ (4) follows from [14; Theorem 2.2] nd [1; Theorem 4]. By Lemm 1.1, we hve (2) ~ (4). (4) ~ (2). Let f: L ~ X be closed s-mp with L loclly compct metric. Since L is determined by loclly finite cover J of compct metric subsets, X is determined by loclly countble cover f(j) of compct metric subsets. Thus by Lemm 2.1, we hve (2). Every chunk-complex is spce dominted by cover of compct metric subsets. However, the uthors do not know whether the converse holds. So, we shll pose the following question.
TOPOLOGY PROCEEDINGS Volume 9 1984 157 Question 2.3. Is every spce dominted by cover of compct metric subsets chunk-complex? Concerning the bove question, if the cover is pointcountble (equivlently, loclly countble), then the nswer is ffirmtive by Lemm 2.1. Among Frechet spces, the nswer is lso ffirmtive without the point-countbleness, nd in ddition, every Frechet chunk-complex is chrcterized s the closed imge of loclly compct metric spce. To prove this, we need the following lemm. Lemm 2.4. Let X be Frechet spce dominted by cover {X ; < B} of compct subsets, nd let Y = X - Uo<Xo. Then for ech, there exists finite subset I such tht U{Y n Y ; 0 I} is finite. o Proof For some ' suppose tht U{Y n Y ; I} o o is infinite for ny finite subset I. Then, by induction we cn choose countble distinct points Yn E X nd countbly mny ordinls n with n < + l such tht Yn E Y n Y n n o for ech n E N. Since Yn E Y ' there exists sequence n L n in X - Uo< Xo converging to the point Y n. Since n n the points Y re contined in compct subset X ' there n o exists s eque nce {yn. ' i E N} ccumulting to point 1 y E X with y Then, ṿ 1 U{L 0 n. ; i EN}. Thus o n. o J.. 1 there exists sequence P = {p. ; j E N} converging to the J point Yo such tht p. E L nd Pj ~ Yo But, P n X J ni(j) ni(j) is finite for ech j E N, so tht P is closed in X. Hence, Yo E P, contrdiction.
158 Tnk nd Ho-xun Theorem 2.5. The following re equivlent. (1) X is Frechet spce dominted by cover of compct metric subsets. spce. (2) X is Frechet chunk-complex. (3) X is the closed imge of loclly compct metric Proof (2) ~ (3) follows from Theorem 1.2. (3) ~ (1). The closed imge X of loclly compct 4 metric spce hs hereditrily closure-preserving cover of compct metric subsets. Then X is dominted by this cover. (1) ~ (2). To show tht X is chunk-complex, let J = {Y ' < B} be collection of the subsets Y in Lemm 2.4; here the Y re compct metric. Let H be the cover consisting of ll finite intersections of members of J. We will prove tht H is cover of chunks for X. Since X is Frechet, in view of Lemm 1.1, J is hereditrily closure-preserving cover of X. Thus X is determined by J. Since J c H, X is determined by H. To show tht for ech S E H, {T E H; T c S} is finite, tke ny Y E J with o S c Y By Lemm 2.4, there exists finite subset 1 0 o such tht F U{Y n Y ; 1 } is finite. For ny 0 o 0 T c S, let T n{t. ; i 1,2,,n}. Then ll. E I 1 0' 1 otherwise some. I. The ltter cse implies tht 1 0 0 n Y c F ' so tht T is subset of the finite o o 4A cover J = {F; E A} of spce is hereditrily closure-preserving if U{A : E A} = U{A : E A} for ny A C F.
TOPOLOGY PROCEEDINGS Volume 9 1984 159 set F o. Hence, {T E H; T cs} is finite, becuse there exist only finitely mny subsets of the finite set 1 only finitely mny subsets of the finite set F. o 0 nd Let f: X ~ Y be closed mp such tht X is dominted by closed cover J. Then it is esy to show tht Y is dominted by f(j). Thus we hve the following by Theorem 2.5. Corollry 2.6. Every Frechet spce which is the closed imge of chunk-complex is chunk-complex. Corollry 2.7. Every Frechet spce dominted by closed cover of chunk-complexes is chunk-complex. Proof. Let X be Frechet spce dominted by closed cover {X ; < S} of chunk-complexes, nd let Y = X - Uo<Xo. Then by Lemm 1.1, X is the closed imge of the topologicl sum of Y's. of chunk-complex X But ech closed subset Y is dominted by compct metric sub sets. Since Y is Frechet, by Theorem 1.2, Y is the closed imge of loclly compct metric spce, hence so is X. Thus X is chunk-complex by Theorem 2.5. 3. CW-Compleses Let X be complex. A subcomplex L of X is the union of subset of the cells of X, which re the cells of L, such tht, if eel then eel. Recll tht X is CW-compZex if it is dominted by the cover of ll finite subcomplexes, nd ech cell is contined in finite subcomplex. Note tht every CW-complex is determined by the
160 Tnk nd Ho-xun cover of ll closed cells e, but not lwys dominted by this cover. Now, every ew-complex is chunk-complex. Thus, by Theorem 2.5, every Frechet ew-complex is the closed imge of loclly compct metric spce. But this conclusion cn be refined by writing "ew-complex" insted of "spce." To show this, we need the following lemm. The proof is essentilly the sme s in the proof of Lemm 2.4. Lemm 3.1. Let X be ew-complex with cells {e}. If X is Frechet, then for ech e E X there is finite collection E of cells such tht u{ n ei E X - E} is finite. Theorem 3.2. Let X be ew-complex. Then the following re equivlent. (1) X is Frechet spce (resp. Frechet, loclly 5 countble ew-complex ). (2) X is the closed imge (resp. closed s-imge) of loclly compct metric ew-complex. the topologicl sum of Eucliden simplexes.) (The domin is ctully Proof. Since every Lsnev spce is Frechet, (2) ~ (1) is obvious. For the prenthetic prt, since X is loclly seprble spce, the ew-complex X is loclly countble. To prove (1) ~ (2), let X be Frechet ew-complex with cells {e}. Let X* be the topologicl sum Le* of e's. Since ech e* is the closed imge of Eucliden simplex 0, SA ew-complex X is loclly countble, if for ech x E X there exists countble subcomplex A of X such tht x E int Ai equivlently, the cover of ll closed cells of X is loclly countble.
TOPOLOGY PROCEEDINGS Volume 9 1984 161 x* is the closed imge of Io. Thus it suffices to prove tht the obvious mp f: X* ~ X is closed. Let A be closed subset of X*. Then we will prove tht f(a) n is closed in X for ech cell d E X, so tht f(a) is closed in X. For d E X, let D be finite subcomplex of X contining d. Let A = f(a n e~) n d for ech e E X. Then, e f(a) n d = U{A e E X} = (U{A ; e E D}) U (U{A ; e D}). e' e e Since ech A is closed in X, so is U{A e; e E D}. Thus it e is sufficient to prove tht B = U{A e; e D} is closed in X. By Lemm 3.1 there is finite collection F of cells such tht C = U{e n d'; d' E D, e E X - F} is finite. Since B U{A n d' ; d' E D, e D}, B = (U {A n d' ; e e d' E D, e E F - D} ) U (U {A n d' ; d' E D, e E X - (F U D) } e Since U{A n d' ; d' E D} = A is closed in X, so is the e e first union. The second union is contined in finite subset C n d = U{ (e n d') n d; d' E D, e E X - F}. Thus the second union is finite, hence is closed in X. Thus B is closed in X. Hence f is closed mp. Tht completes the proof. The proof of the previous theorem suggests the "only if" prt of the following. The "if" prt is obvious. Corollry 3.3. Let X be CW-complex with cells {e}. Then X is Frechet if nd only if {e} is hereditrily closure-preserving cover of x. Note Added in Proof. The uthors hve recently shown tht every Frechet spce dominted by closed cover of metric subsets is decomposed into metric subset nd closed discrete subset.
162 Tnk nd Ho-xun References v [1] A. V. Arhnge1'skii, Some types of fctor mppings nd the reltions between clsses of topologicl spces, Soviet Mth. Dok1. 4 (1963),1726-1729. [2] J. G. Ceder, Some g~nerliztions of metric spces, Pcific J. Mth. 4 (1961), 105-126. [3] C. H. Dowker, Topology of metric complexes, AIDer. J. Mth. 74 (1952), 557-577. [4] R. Engelking, Generl Topology, Polish Scientific Publishers, Wrszw, 1977. [5] S. P. Frnklin nd B. V. Smith Thoms, On the metrizbility of kw-spces, Pcific J. Mth. 72 (1977), 399 402. [6] G. Gruenhge nd Y. Tnk, Products of k-spces nd spces of countble tightness, Trnsctions of AIDer. Mth. Soc. 273 (1982), 299-308. [7] G. Gruenhge, E. Michel nd Y. Tnk, Spces determined by point-countble covers, Pcific J. Mth. 113 (1984), 303-332. [8] N. Lsnev, Continuous decompositions nd closed mppings of metric spces, Soviet Mth. Dok1. 165 (1965), 1504 1506. [9] E. Michel, A quintuple quotient quest, Generl Topology nd App1. 2 (1972), 91-138. [10] K. Morit, On spces hving the wek topology with respect to closed coverings, Proc. Jpn Acd. 29 (1954), 537-543. [11], On closed mppings, Proc. Jpn Acd. 32 (1956), 539-543. [12] Y. Tnk, Products of sequentil spces, Proc. AIDer. Mth. Soc. 54 (1976), 371-375. [13], Point-countble k-systems nd products of k-spces, Pcific J. Mth. 101 (1982), 199-208. [14], Closed imges of loclly compct spces nd Frechet spces, Topology Proceedings 7 (1982), 279-292. [15], Metrizbility of certin quotient spces, Fund. Mth. 119 (1983), 157-168.
TOPOLOGY PROCEEDINGS Volume 9 1984 163 [16] J. H. C. Whitehed, Combintoril homotopy. I, Bull. Amer. Mth. Soc. 55 (1949), 213-245. Tokyo Gkugei University Kognei-shi, Tokyo, Jpn nd Sichun University Chengdu, Chin (P.R.C.)