MORE ON CAUCHY CONDITIONS

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1 Volume 9, 1984 Pages MORE ON CAUCHY CONDITIONS by S. W. Davis Topology Proceedigs Web: Mail: Topology Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume MORE ON CAUCHY CONDITIONS s. W. Davis o. Itroductio This ote is a cotiuatio of [0]. The questio \oj which prompts this wor was ased by Arhagel'sii ad relayed by M. E. Rudi, amely, "Is every Lidelof symmetrizable space separable?" This questio appears i prit i [NC] The approach which has bee used o this problem by Kofer [K], Nedev [N] ad this author has bee to icrease the stregth of the symmetric ivolved by addig a Cauchylie requiremet for coverget sequeces. These coditios are described i detail i [0], ad we will ot repeat them here with the exceptio of the coditio which we will be usig. Defiitio 0.1 [A]. A space X is called wealy first coutable iff there exists B: w x X + ~(X) such that (i) for all x E X we have B( + l,x) ~ B(,x) for all Ew, ad EwB(,x) = {x} ad (2) a set U ~ X is ope iff for each x E U there exists E w such that B(x'x) ~ U. x The fuctio B i 0.1 is called a wfc-system which is compatible with X. Defiitio 0.2. If B is awfc-system for a space X, the we say B is wc iff wheever A c X ad there exists Ew such that B(,x) A = {x} for all x E A, the A is relatively discrete.

3 32 Davis The preset paper cocers wc wfc-systems o symmetrizable spaces. We show that if a Lidelof symmetrizable space has a wc wfc-system, the it is separable. We also preset a example of a symretrizable space with o wc wfc-system which aswers a questio raised i [D]. We use the otatio w = {O,1,2, } ad ~ {1,2, }. 1. The Mai Result I this sectio, we give the (surprisigly easy) proof of our mai theorem. First, we give a lemma which is quite useful i worig o these properties. Lemma 1.1. Suppose X is a T 2 -space, d is a symmetric compatible with' X, ad B is a wfc-system compatible with X. For each x E X ad E N, there exists E w such that 1 B(,x) ={y: d(x,y) < }. Proof. If the theorem is false, the choose x E X ad E N witessig that fact. For each E w choose x E B(,x) with d(x,x ) ~~. Sice B is compatible with x, the sequece (x : E w) coverges to x. Co;sider the set {x : E w}. Suppose y {X : E w}. If Y = x, the d(y,{x : E w}) ~ ~ > O. If Y ~ x, the {x} U {x : E w} is a closed set excludig y, thus d(y,{x : E w}) ~ d(y,{x} U {x : E w}) > O. I either case the d(y,{x : E w}) > 0 for all y E X\{x : E w}. Sice d is compatible with X, this says that {x : E w} is closed which is clearly impossible sice x E {x : E w}\{x : E w}. Hece we have the result.

4 TOPOLOGY PROCEEDINGS Volume Remar. Although we shall ot eed it here, a somewhat stroger result ca be obtaied; amely, if X is T 2 ad B, B' are wfc-systems compatible with X, the for each x E X ad E N, there exists EN such that B(,x) c B' (,x). Lemma 1.2 [Nedev]. Every regular Lidel8f symmetrizable space is hereditarily Lidelof. Besides beig useful for the ext result, 1.2 shows that a couterexample (if there is oe) to the mai questio must be a L-space. Theorem 1.3. If X is a regular Lidelof symmetrizable space with a compatible wc wfc-system, the X is separable. Proof. Let d be a symmetric compatible with X, ad B a wc wfc-system compatible with X. For each E N, let 1 D ~ X be maximal with respect to d(x,y) ~ for x,y E D ' x ':I y. Now D 00 U=lD is dese i X, sice if x E U with U ope ad U D = ~, the choose E N with d(x,x\u) > 1 ad D is ot maximal. Fix E N. For each ' x E D ' by Lemma 1.1, we let (x,) be a elemet of the set {: B(,x) c {y: d(x,y) <!}}. For each E w, let - E {x E D : (x,) = }. For each x E E, B(,x) E {x}. Sice B is wc, E is the relatively discrete. By Lemma 1.2, X is hereditarily Lidelof, so E is coutable. Now sice D = UEwE, we have that D is coutable. This is true for each E N, thus D is the desired coutable dese subset.

5 34 Davis 2. The Mai Example I this sectio, we describe a old example of Kofer [K], ad show that it has o compatible we wfc-system eve though it is a separable symmetrizable space. This aswers egatively Questio 4.1 of [D]. Example 2.1 [Kofer]. Let R deote the real umbers, I deote the irratioal umbers, ad Z deote the itegers. Let DO = ~, ad for i E N, let D. = {~l: E Z}. Let D = UiEwD i Note that D is the set of dyadic ratioals. Let M = I U D. For x E M ad j E N, we let T. (x) = M J 1 1 (x - -'-1' x ) For x E I, let S. (x) = T. (x) D. J- J- J J J 2 2 for each j E N. Let O (x) U. S. (x) U {x} for each E N. J~ ] let r For xed, choose i with x E Di\D i - For E N, l Tr(X) U U(x). rax{i,}, let U(x) = T(X) D, ad let 0(x) Defie B*: N x M + ~(M) by B*(,x) = 0(x). The usig (2) of Defiitio 0.1 to defie a topology o M, B* is a wfc-syster. Let X be the space whose uderlyig set is M with the topology geerated by B*. Lemma 2.2 [Kofer]. B* is a symmetric wfc-system (i.e. x E B*(,y) ~ y E B*(,x)) for the topology o X which is fier tha Euclidea. Hece X is T ad symmetriza 2 ble. Lemma 2.3 [Kofer]. X is zero dimesioaz. Hece X is a Tychooff space.

6 TOPOLOGY PROCEEDINGS Volume Theorem 2.4. There is o wc wfc-system which is compatibze with X. Proof. Suppose B is ay wfc-system which is compatible with X. For Ew, let I = {x E I: B(,x) I = {x}}. Sice for each x E I, there exists E w such that B(,x) ~ 0l(x) by Lemma 1.1, we have that UEwI = I. Sice I is secod category i R, choose Ew ad a ope iterval (a,b) such that a ~ M, b ~ M, ad I (a,b) is dese i the Euclidea topology o (a,b). Now i the topology of X, each poit of I (a,b) is a cluster poit of I so that I ca ot be relatively discrete. Hece B is ot wc. 3. Sequetial Separability ad Questios The positive results o the questio we've bee cosiderig all have a similarity i the way the coutable dese subset is costructed. Ideed, the coutable dese set D is such that each poit i the space is the limit of a sequece draw from D. I o-frechet spaces, this "sequetial separability" seems 'too much to expect. Questio 3.1. "sequetially separable"? * Is every separable symmetrizable space The mai questio could still be aswered by a affirmative aswer to the followig: Questio 3.2. Does every Lidelof symmetrizable space admit a wc wfc-system? * See Note added i proof.

7 36 Davis Related to wc ad wea Cauchy coditios (see [D] for defiitios), the followig questios remai. Questio 3.3. If X has a symmetric ad a wc wfc-system, must X have a wc symmetric? Questio 3.4. If X has a wc symmetric, must X have a wealy Cauchy symmetric? Kofer [K] has show that the aswer is "yes" to 3.3 if "wc" is replaced by "wealy Cauchy." Example 2.5 of [D] shows that 3.4 does ot trivially have a "yes" aswer. Note (added i proof). L. Foged has give a example aswerig 3.1 i the' egative. Refereces [A] A. V. Arhagel'sii, Mappigs ad spaces, Russia Math. Surveys 21 (1966), [D] S. W. Davis, Cauchy coditios o symmetrics, Proc. Amer. Math. Soc. 86 (1982), [K] J. A. Kofer, SymmetpizabZe spaees ad faetop mappigs, traslated from Matematichesie Zameti 14 (1973), [N] S. I. Nedev, Symmetrizable spaces ad fial. compactess, Soviet Math. Dol. 8 (1967), [NC] ad M. M. Choba, O the theory of o-metrizable spaces, II, Vesti Mosovsogo Uiversiteta. Matematia, 27 (2) (1972), Miami Uiversity Oxford, Ohio 45056