SYMMETRIZABLE SPACES AND SEPARABILITY

Transcription

1 Volume 4, 1979 Pges SYMMETRIZABLE SPACES AND SEPARABILITY by R. M. Stephenson, Jr. Topology Proceedings Web: Mil: Topology Proceedings Deprtment of Mthemtics & Sttistics Auburn University, Albm 36849, USA E-mil: ISSN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume SYMMETRIZABLE SPACES AND SEPARABILITY R. M. Stephenson, Jr. For severl yers topologists hve been interested in discovering properties of feebly compct syrnmetrizble spces nd in lerning if they must ll be seprble nd e-countbly compct (defined below), s is the cse with the well-known feebly compct Moore spce ~ in [GJ, 51] (see lso [M3, p. 66 nd p. 381]). Some of the recent results which hve been obtined will be given here, nd exmples will be constructed in order to prove tht (1) there exists feebly compct Moore spce which is not e-countbly compct, nd (2) there exists feebly compct semimetrizble Husdorff spce which is not seprble. Recll tht topologicl spce is clled feebly compct if every loclly finite system of open sets is finite. It is well-known tht in completely regulr spces feeble compctness nd pseudocompctness (every continuous rel vlued function is bounded) re equivlent concepts, nd in norml spces feeble compctness,nd countble compctness re equivlent. A topologicl spce X is sid to be symmetrizble [All if there exists mpping d: X x X ~ [0,00) such tht: i) d(x,y) d(y,x) for ll x,y E Xi ii) d(x,y) 0 i-f nd only if x = Yi nd iii) for every set V c X, V is open if nd only if for ech point v E V, there exists e > 0 such tht V => B(v,e) - {x: d(x,v) < e}.

3 590 Stephenson In cse d cn be chosen so tht, in ddition, ech B(v,e) is neighborhood of v, then X is clled semimetrizble. Let us first consider seprbility nd then e-countble compctness. 1. Seprbility nd Feeble Compctness The first result I know of linking these concepts for symmetrizble spces is the following. Theorem 1.1 (Reed [R]). Every Moore-closed spce is seprble. Recll tht if P is property of topologies, P-spce X is clled P-closed if X is closed subspce of every P-spce in which it cn be embedded. By result of J. Green [G] Moore spce is Mooreclosed if nd only if it is feebly compct. Severl yers fter Theorem 1.1 ws discovered, R. W. Heth obtined the following result. Theorem 1.2 (Heth). Every regulr, feebly compct semimetrizble spce is Moore spce. Thus, the hypothesis of 1.1 could be t lest formlly wekened to regulr, feebly compct, nd semimetrizble. Then in 1977 the next result ws obtined. Theorem 1.3 ([52]). Every Bire, feebly compct semimetrizble spce is seprble. Becuse every regulr, feebly cpmpct spce is Bire

4 TOPOLOGY PROCEEDINGS Volume ([M2], see lso [CN]), Theorem 1.3 is n extension of Theorem 1.1. A construction given in [53] showed tht "semimetrizble" could not be replced by "symmetrizble" in 1.3. It ws proved there tht for every crdinl number n, there exists Bire, feebly compct, symmetrizble Husdorff spce hving no dense subset of crdinlity ~ n. One direction in which 1.3 cn be extended is indicted by the following result concerning the fmily J studied in [D], [DGN], [DS], nd [HS]. Theorem 1.4 ([DS]). Let X be Bire, feebly compct neighborhood J-spce, nd let I be the set of isolted points of x. Then X hs dense subset D with IDI < mx{ii!,h }. - 0 We will now give n exmple which provides negtive nswer to the question [52]: Is every feebly compct semimetrizble spce seprble? Theorem 1.5. Let n be n infinite crdinl number. Then there exists Husdorff, feebly compct developble spce X which hs no dense subset of crdinlity < n. Construction. Let Q be the set of rtionl numbers, with its usul topology, let the crdinl number n hve the discrete topology, nd let Y = Q x n hve the product topology. Let N denote the set of nturl numbers nd list the members of Q s {qi: i EN}. For ech i E N let F i = {qk: k < i} x n. Let 5 be the set of ll countble clopen filter bses V on Y such tht: i) V hs no dherent point in Y, i.e., n {V: V E V} $1 nd

5 592 Stephenson ii) the sets comprising V cn be lbeled {Vi: i EN}, where for ech i, V. ~ V. 1 nd V. n F. =~. Next, let T be mximl subset of S such tht whenever V,W E! with V ~ W, then there exist disjoint sets V E V nd W E W. Select distinct points PV f. Y for ech V E!, nd let X = Y U {PV: V E!}, topologized s follows: ech open subset of Y is open in Xi nd neighborhood of pointp V is ny set U c X such tht for some V E V, one hs U ~ {PV} u V. Proof. Becuse no merr~er of T hs n dherent point in Y, nd becuse two distinct members of T contin t lest two disjoint sets, it is esy to see tht X is Husdorff spce. A development for X will now be given. For ech i E N, men, nd point qk E Q let { (q, m) E Y: Iq - qk I < 1/i} Lbel the members of members of T s in ii) nd for ech i E N nd VET let B (PV' i) = {PV} u v 1. Define lj. = {B ( (qk ' m), i) : (qk,m) E Y)} U {B (PV' i) : V E!}, 1 i E N. Tht lj., i E N, is development for the spce X is 1 cler except possibly t the points of Y. For point (qk,m) nd n open neighborhood U of (qk,m), find j E N with B(qk,m),j) c U, nd note tht for i > mx{2.,k}, one hs - J str «qk,m),lj i ) c U, since (qk,m) E F nd hence (qk,m) ~ i B(PV,i) for ny V E!. Suppose X fils to be feebly compct. Then [BCM] there exists pirwise disjoint, loclly finite, countbly

6 TOPOLOGY PROCEEDINGS Volume infinite fmily {U : i E N} of nonempty open subsets of X. i For ech i E N, F i is closed nowhere dense subset of Y nd Y is dense subset of X, so there exists nonempty clopen subset C. of Y with C. c U.\F. Let V ' = U{C : k > i} nd k V = {V.: i E N} nd note tht V hs no dherent point in X 1 since {C : ken} is loclly finite. But clerly i) nd ii) k re stisfied nd so V E S. By the mximlity of!, there must exist WE! such tht V n w ~ ~ for ll sets V E V nd W E W. From the ltter, however, it would follow tht Pw is n dherent point of V, in contrdiction of the fct tht V hs no dherent point in X. To complete the proof, it suffices to observe tht Y is n open subset of X hving no dense subset of crdinlity < n, nd so X cnnot hve dense subset of crdinlity < n. Let us conclude this section with question. Question 1.6. Is ~very regulr, feebly compct symmetrizble spce seprble? As noted in [S3], n exmple providing negtive nswer to 1.6 would lso provide negtive nswer to the question of E. Michel: Is every point of regulr symmetrizble spce Go? (Becuse, by result of I. Glicksberg, every Go point of regulr feebly compct spce hs countble neighborhood bse, nd it is known tht Husdorff first countble symmetrizble spce is ctully semimetrizble.). The reder interested in countble chin nd completeness type xioms which imply tht Moore spce be seprble is referred to [A2], [H], [Ml], [M4], nd [R].

7 594 Stephenson 2. e-countble'compctness nd Feeble Compctness A topologicl spce X is clled e-countbly compct with respect to dense subset D if every infinite subset of D hs limit point in X. A spce X is clled e-countbly compct if there exists dense subset of X with respect to which X is e-countbly compct. Like every feebly compct spce hving dense set of isolted points, the noncompct Moore-closed spce ~ of [GJ, 5I] must be e-countbly compct (with respect to its set of isolted points). An interesting question rised by J. Green is the following. Question 2.1 (Green [G]). Does every noncompct Moore-closed spce contin noncompct 3 e-countbly compct subspce? The construction below provides prtil nswer Exmple 2.2. There exists loclly compct, zerodimensionl Moore-closed spce X which fils to be e-countbly compct. Description of X. Let C be the Cntor set, N = the set of nturl numbers, nd Y = C x N, with the product topology. Let c = 2~o nd {M : < c} be 1-1 listing of the members of mximl infinite fmily mof lmost disjoint infinite subsets of N. Let 0 = {D : < c} be the fmily of ll countble dense subsets of Y, nd for ech ordinl < c nd nturl number n E M, choose one point nd let d,n E D n (C x {n}),

8 TOPOLOGY PROCEEDINGS Volume J {d n EM}.,n Next, for ech < c, let! be mximl fmily such tht: i) ech] E T is countbly infinite, pirwise dis - joint, loclly finite collection of compct open subsets of C x M ; ii) if ]1 nd ]2 E! nd ]1 ~ finite sets ]i c J i, (U(]2']2)) = ~; nd ]2' then there exist i = 1,2, so tht (U(]l']l)) n iii) ech] E T stisfies J n (U]) =~. Finlly, --. select distinct points PJ~, t Y nd let X = Y U {PJ~ : < c nd] E T },, --. topologized s follows: ech open subset of Y is open in X; nd neighborhood of point PJ~ is, ny set V C X such tht for some finite set] c ], one hs V~{PJ~ } U (U(]V)), Proof. To see tht X is Husdorff, consider distinct points x,y E X. If x,y E Y, it follows from the openness of Y in X tht x,y hve disjoint neighborhoods. Suppose x E Y nd y = P],; then by the locl finiteness of J in Y there exist n open neighborhood V of x in Y nd finite subset ] of ] such tht V n ({PJ,} U (U(J\]))) =~. If x = p], nd y = PV, ' where ] ~ V, then by ii), one cn esily find disjoint neighborhoods of x nd y. Finlly, consider the cse x = p], nd y = PV,b' where ~ b. The set F = M n M is finite, so C x F is compct nd there must exist b finite subsets] of ] nd ~ of V such tht (U (]\])) n (C x F) = <p = (U (V\~)) n (C x F).

9 596 Stephenson Then {x} U (U(],]» nd {y} U (u(v'9) re disjoint neighborhoods of x nd y. Since Y is loclly compct nd zero-dimensionl subspce of the Husdorff spce X, nd since ech point of X\Y clerly hs neighborhood bse consisting of compct open subsets of X, the spce X is loclly compct nd zero-dimensionl. A development y,n E N, will now be defined for X. n For ech n E N, point (x,k) E Y, nd point p], E X\Y, let nd B ( (x, k), n) = {(y,k) E Y: Ix - y I < ~} B (pjr-:,,n) {P],} U (U{T E ]: T n (c x {1,2,,n}) <I> } ) It suffices to tke, for ech n E N, y n = {B ( (x,k), n): (x, k) E Y} U {B (pjr-:,, n) : pjr-:, E X\Y}. To verify tht X fils to be e-countbly compct, note tht if D is ny dense subset of X, then D n Y is dense subset of the second countble spce y (since y is open in X) nd so for some < c., D c: D. Then J is n infinite subset of D which by i) nd iii) hs no limit point in X. To complete the proof, suppose there exists n infinite loclly finite fmily C of open subsets of X. Then, becuse Y is dense in X, one cn find set M E m, n infinite set H c: M, nd 1-1 mpping f: H ~ C such tht for ech n E H, (C x {n}) n f(n) ~ <1>. Further, for ech n E H one cn find compct open set K c: (C x {n}) n (f(n)\{d }). Then n,n K = {K : n E H} is countbly infinite, pirwise disjoint n fmily of compct open subsets of (C x M)\J which is

10 TOPOLOGY PROCEEDINGS Volume loclly finite in X nd Y. But then, by the mxim1ity of T, there must exist JET such tht for every finite sub - -- set] of ], one hs (U{]\]» n K # for infinitely mny n n E H. Thus K nd, hence, C fil to be loclly finite t the point PJ~,which is contrdiction., We will conclude by stting some results which relte e-countb1e compctness nd seprbility. Theorem 2.3 ([Sl]). (i) Every e-countbly compct semimetrizble spce is seprble. (ii) If symmetrizble spce is e-countbly compct with respect to set D, then every discrete subspce of D is countble. Theorem 2.4 (Nedev [N]). Every countbly compct (Husdorff) symmetrizble spce is compct (nd metrizble). Question 2.5. Is every e-countbly compct symmetrizble spce seprble? References [A1] [A2] [BCM] A. V. Arhnge1'skii, "Mppings nd spces, Russin Mth. Surveys 21 (1966), S. Armentrout, Completing Moore spces, Proc. Top. Conf. Arizon Stte University (1967), R. W. Bgley, E. H. Connell, nd J. D. McKnight, Jr., On properties chrcterizing pseudocompct spces, Proc. Arner. Mth. Soc. 9 (1958), 500~506. [en] W. W. ~omfort nd S. Negrepontis, Continuous pseudometrics, Lecture Notes in Pure nd Applied Mthemtics, Vol. 14 (1975), Mrcel Dekker, New York. [D] S. W. Dvis, On ] -spces, Generl Topology nd Its r App1.9 (1978),

11 598 Stephenson [DGN] [DS] [GJ] [G] [H] [HS] [MIl [M2l [M3] [M4] [N] [R] [Sl] S. W. Dvis, G. Gruenhge, nd P. J. Nyikos, Go-sets in symmetrizble nd relted spces, Generl Topology nd Its App1. 9 (1978), S. W. Dvis nd R. M. Stephenson, Jr., Seprbility nd miniml wek bse topologies, Proc. Amer. Mth. Soc. 74 (1979), L. Gillmn nd M. Jerison, Rings of continuous functions, Vn Nostrnd, Princeton, N.J. (1960). J. W. Green, Moore-closed spces, completeness, nd centered bses, Generl Topology nd Its App1. 4 (1974), R. W. Heth, Semi-metric spces nd relted spces, Proc. Top. Conf. Arizon Stte University (1967), P. W. Hrley III nd R. M. Stephenson, Jr., Symmetrizble nd relted spces, Trns. Amer. Mth. Soc. 219 (1976), L. F. McAuley, A reltion between seprbility, completeness nd normlity in semi-metric spces, Pcific J. Mth. 6 (1956), R. A. McCoy, A filter chrcteriztion of regulr Bire spces, Proc. Amer. Mth. Soc. 40 (1973), R. L. Moore, Foundtions of point set theory, Amer. Mth. Soc. Colloq. Pub1. 13 (1932)., Concerning seprbility, Proc. Nt. Acd. Sci. U.S.A. 28 (1942), Nedev, Symmetrizble spces nd finl compctness, Soviet Mth. Dok1 8 (1967), G. M. Reed, On chin conditions in Moore spces, Generl Topology nd Its App1. 4 (1974), R. M. Stephenson, Jr., Symmetrizble, J-, nd wekly first countble spces, Cn. J. Mth. 29 (1977), [S2], Symmetrizble-closed spces, Pcific J. Mth. 68 (1977), [53], Ner compctness nd seprbility of symmetrizble spces, Proc. Amer. Mth. Soc. 68 (1978),

12 TOPOLOGY PROCEEDINGS Volume University of South Crolin Columbi, South Crolin 29208