STRONG QUASI-COMPLETE SPACES

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1 Volume 1, 1976 Pages STRONG QUASI-COMPLETE SPACES by Raymod F. Gittigs 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 STRONG QUASI-COMPLETE SPACES Raymod F. Gittigs 1. Itroductio May cocepts i metrizatio theory have either bee defied or ca be characterized by meas of a sequece of ope covers which guaratee that certai sequeces have cluster poits. These cocepts actually occur i upairs u with the stroger cocept requirig a certai type of sequece to cluster at a particular poit, whereas the weaker cocept merely requires a sequece of the same sort to cluster. The purpose of this paper will be to itroduce a class of spaces motivated by this "pair u occurrece, ad to ivestigate the relatioship to various other importat classes of spaces. Uless otherwise stated, o separatio axioms are assumed; however, all regular spaces are assumed to be T The positive l. itegers are deoted by N. If GU is a cover of X the GU * = {St(uflL): U EGlL}. Let <GU> be a sequece of ope covers of a space X. Cosider the followig coditios o the sequece <Gl.J~ >. (A) (i) GU 1 >GU; >GU >GU; > 2 (ii) If x E St (x,gll ), the the sequece < x > has a cluster poit. (B) If x E St 2 (x,gll ), the the sequece < x > has a cluster poit. poit. (C) If x E St (x,gll ), the the sequece < x > has a cluster (D) If {xi: i ~ } U {x} C U E GU' the the sequece < x > has a cluster poit. * Supported by the Research Foudatio of the City Uiversity of New York, Grat No

3 244 Gittigs A space X with a sequece <GU> satisfyig (A), (B), (C) or (0) is called a M-space [20], wm-space [18], w~-space [4] or a quasi-complete space [10], respectively. If, i (A), (B) ad (C), we require that the sequece <X > clusters to x, the (A) ad (B) are well-kow characterizatios of metrizability (at least for TO-spaces) ad (C) is clearly equivalet to the defiitio of a developable space. I fact, (A) is the coditio of the Alexadroff-Urysoh Metrizatio Theorem [1], ad (B) is that of the Moore Metrizatio Theorem [19] (see also [18, Theorem 2.3 i II]). The coditios give i (A), (B) ad (C) illustrate our previous discussio cocerig certai cocepts occurrig i "pairs," ad motivate the followig defiitio: A space X is called a strog quasi-complete space if there exists a sequece <G\L > of ope covers of X such that if {xi: i > } U {x} C the the sequece < x > clusters to x. U E GU' The sequece <~> will be called a strog quasi-complete sequece. I f the sequece <GU > satisfies coditio (0) it will be called a quasi-complete sequece. The basic implicatios amog the cocepts defied i (A), (B), (C) ad (0) are give i the followig diagram: M-space J wm-space J metrizable w~-space developable 1 quasi-complete ~ strog quasi-complete Noe of the implicatios are reversible; moreover, the space [O,~), where ~ is the first ucoutable ordial, is a M-space which is ot strog quasi-complete. A example of a strog quasi-complete space which is ot a developable space will be preseted i Sectio 2 (see Bxample 2.4).

4 TOPOLOGY PROCEEDINGS Volume A atural questio to ask at this poit is the followig: Uder what coditios does a cocept i the first colum imply the correspodig cocept i the secod colum? Several solutios are kow ad these will be discussed i Sectio 2; however, several questios remai ope. 2. Strog Quasi-Complete Spaces I this sectio we discuss the relatioship of strog quasi-complete spaces to other classes of spaces, ad determie whe a quasi-complete space is strog quasi-complete. I [8], Chaber proved the followig importat result. Theorem 2.1 (Chaber [8]) A T 2 -space is metrizable if ad oly if it is a M-space with a Go-diagoal. I light of Chaber's result, the followig problems become particularly iterestig. (1) Is every regular wm-space with a Go-diagoal metrizable? (2) Is every regular w~-space with a Go-diagoal developable? (3) Is every regular quasi-complete space with a Go-diagoal strog quasi complete? I Theorem 2.2 we show that questio (3) has a positive aswer. This is particularly iterestig sice questios (1) ad (2) remai ope eve if we assume collectiowise ormality. Several partial solutios of questios (1) ad (2) are kow. For example, it follows from results i [17] that positive solutios are obtaied if we assume e-refiability or if we replace Go-diagoal by G;-diagoal or o#-space (= a-space [17, Lemma 4.8]). A space X is called a p-space if there exists a sequece < > of ope covers of X satisfyig: If x E X ad G E such that x E G, the (a) 00 G is compact; =l (b) if x E i=lḡ i, t he t h e sequece < x > c 1usters.

5 246 Gittigs The class of p-spaces was itroduced by Arhagel'skiI [2]; however, the above defiitio is the characterizatio obtaied by Burke [5] (complete regularity is ot assumed i our defiitio). Recall that for regular spaces, coditio (b) is a characterizatio of quasi-complete spaces [15]. Accordig to Ceder [7], a space X has a Go-diagoal if ad oly if there is a sequece < > of ope covers of X such that ~=l St (x,g ) = {x} for every x E X. called a Go-diagoal sequece. The sequece < > will be Theorem 2.2. For a regular space X, the followig are (a) X is a strog quasi-complete space. (b) X is a p-space with a Go-diagoal. (c) X is a quasi-complete space with a Go-diagoal. Proof (a) ~ (b): Sice every regular quasi-complete space with a Go-diagoal is a p-space [15, Theorem 3.6], it suffices to show that a strog quasi-complete space has a G o diagoal. Let <~ > be a strog quasi-complete sequece for X. Suppose x, y E X with x ~ y. If Y E ~=lst(xm)' the there exists a U E qr such that {x,y} C U for every E N. It follows that the costat sequece <y> must cluster to x. Sice this is impossible, ~> is a Go-diagoal sequece for X. The fact that (b) ~ (c) is a cosequece of [15, Lemma 3.3]. (c) ~ (a): Let <~> be a quasi-complete sequece for X, ad let < > be a Go-diagoal sequece for X. For each E N, let ~ =~ 1\ ~ = {u G: U E ~, G E }. By regularity, there is a ope coverw 1 of X such that {W: W E WI} <l~l ad, for each 2:. 2, a ope coverlt) of X such that {W: W Elf) } <1'1 I\lf) 1. - We ote that <~> is both a quasi-complete sequece ad a G o diagoal sequece for X. If {Xi: i>} U {x} CWEW, the

6 TOPOLOGY PROCEEDINGS Volume the sequece < x > has a cluster poit y. Sice x E W for k every k 2:., y rt. X - W for ay E N. Hece y E ~=:lw c complete sequece. {x} Thus y = x ad <It) > is a strogr quasi A base of coutabze order [22] for a space X is a base ffi such that if e is a perfectly decreasig subcollectio of ffi (i.e. e cotais a proper subset of each of its members) ad x E {C: C E e}, the e is a local base at x. Theorem 2.3. Every regular streg quasi-complete space X has a base of coutabze order. Proof. Let <GU> be a strog quasi-complete sequece for X. For each E N, let'l:j = {V ope i X: V cue GU } ad ote that'l:j is a base for X. Suppose x E V E 'l:j ad V + c V. l Sice the proof of (a) -+ (b) i Theorem 2.2 shows that <GU> is a GcS-diagoal sequece, it follows easily that ~=lij = {x}. Let W be ay ope set such that x E W. If x E V-W, the {xi: i2:. } U {x} C V because Vi C V if i >. Hece the sequece < x > clusters to x which is a cotradictio. Thus <V > is a local base at x, ad so X has a base of coutable order [22, Theorem 2]. It follows from results i [22], that every regular, e refiable, strog quasi-complete space is developable, ad that every paracompact, strog quasi-complete T -space is 2 metrizable. I order to dispel ay thought o the part of the reader that the cocept of strog quasi-complete might be equivalet to either developable or base of coutable order, we site the followig examples. Example 2.4. A collectiowise ormal, strog quasi-complete space which is ot a developable space.

7 248 Gittigs The space A costructed by va Douwe i [11] is such a space. Sice A is locally compact ad submetrizable, A is a strog quasi-complete space. It follows easily from the fact that A is ormal ad wi-compact, that A is collectiowise ormal. Sice A is ot metrizable, A is ot developable [3, Theorem 10]. The space [O,) shows that a space with a base of coutable order eed ot be strog quasi-complete. Several examples exist which show that strog quasi-complete spaces do ot possess some of the well-kow properties possessed by developable spaces. Recall that every developable Tl-space is 8-refiable [22]; however, the space ~ costructed by va Douwe i [11] is a strog quasi-complete space which is ot eve coutably 8-refiable. Actually, va Douwe shows that ~ is ot coutably metacompact; however, a coutably 8-refiable space is coutably metacompact [14]. The space r of va Douwe ad Wicke [12] is a strog quasi-complete space which is ot eve coutably orthocompact (ote that ~ is orthocompact). As was show earlier, every strog quasi-complete space has a Go-diagoal; however, Burke's Example [6] shows that a completely regular quasi-complete space eed ot have a G 8 diagoal. O the other had, a regular developable space is easily see to have a G 6 -diagoal. I the discussio followig questios (1), (2) ad (3) we oted that replacig Go-diagoal by cr#-space gives a positive aswer to questios (1) ad (2). However, we do ot kow the aswer to the "followig: (4) Is every regular quasi-complete, cr#-space a strog quasi-complete space? 3. Properties of Strog Quasi-Complete Spaces I this sectio we discuss some of the topological properties of strog quasi-complete spaces. Before doig this, however,

8 TOPOLOGY PROCEEDINGS Volume let us give some alterate characterizatios of strog quasicomplete spaces. The proofs will be left to the rea.der. Theorem 3.1. For a space X~ the followig are equivalet: (i) X is a strog quasi-complete space. (ii) There exists a sequece <GU> of ope covers of X such that if x E U E GlL ~ the {U : E N} is a local subbase at x. (iii) There exists a sequece <"ll> of ope covers of X such that if {xi: i 2:. } c U{U: U E GlL~} ad x E {U: U E ('ll~} for some fiite subset"ll' cgu ~ the < x > clusters to x. The equivalece of (i) ad (ii) is easy ad is oted i [13]. That (i) ad (iii) are equivalet follows exactly as i [15, Lemma 3.4]. It is iterestig to ote that if we allow the f iite collectio GU' i (iii) to be coutable, ~~e actually obtai a characterizatio of developable spaces. Strog quasi-complete spaces exhibit much better behavior tha quasi-complete spaces with respect to subspaces ad products. It is kow that quasi-complete spaces are ot hereditary [15] ad ot coutably productive [16]. Theorem 3.2. (a) Every strog quasi-complete space is hereditarily strog quasi-complete. (b) If < X > is a sequece of strog quasi-complete.spaces~ the X = TI~=lX is a strog quasi-complete space. Proof. The result i (a) follows easily from the characterizatio of strog quasi-complete spaces give i Theorem 3.1 (ii). That (b) holds follows from [16, Theorem 3.1]. I [6], Burke shows that the perfect image of a locally compact T -space with a Go-diagoal (hece a strog quasi 2 complete space) eed ot have a Go-diagoal. It follows that strog quasi-complete spaces eed ot be preserved by perfect maps. The space Y of Chaber [9] shows that the ope compact 2

9 250 Gittigs image of completely regular, metacompact, complete Moore space (hece a strog quasi-complete space) eed ot be quasi-complete or have a Go-diagoal. I [21, Example 3.7], Taaka costructs a regular, paracompact space with a Go-diagoal which is ot metrizable, but which is the ope fiite-to-oe preimage of a compact metric space. strog quasi-complete. It follows that Taaka's space is ot The space [O,Q), where Q is the first ucoutable ordial, is a M-space ad thus the quasi-perfect preimage of a metrizable space [20, Theorem 6.1]. However, [O,Q) does ot have a Go-diagoal ad is thus ot strog quasicomplete. Summarizig these results, we have: (1) Strog quasi-complete spaces eed ot be preserved by perfect maps or ope compact maps. (2) The preimage of a strog quasi-complete space uder a ope fiite-to-oe map or a quasi-perfect map eed ot be a strog quasi-complete space. Refereces 1. P. S. Alexadroff ad P. Urysoh, Ue coditio essaire et suffisate pour qu'ue class (L) soit ue class (D), C. R. Acad. Sci. Paris 177 (1923), A. V. Arhagel'skii, O a class of spaces cotaiig all metric spaces ad all locally bicompact spaces, Soviet Math. Dokl. 4 (1963), R. H. Big, Metrizatio of topological spaces, Caad. J. Math. 3 (1951), C. J. R. Borges, O metrizability of topological spaces, Caad. J. Math. 20 (1968), D. K. Burke, O p-spaces ad w6-spaces, Pacific J. Math. 35 (1970), , A odevelopable l~cally compact Hausdorff space with a Go-diagoal, Ge. Topology Appl. 2 (1972), J. G. Ceder, Some geeralizatios of metric spaces, Pacific J. Math. 11 (1961), J. Chaber, Coditios which imply oompaotess i ooutably

10 TOPOLOGY PROCEEDINGS Volume compact spaces, preprit. 9., Metacompactess ad the class MOBI, preprit. 10. G. D. Creede, Cocerig semi-stratifiable spaces, Pacific J. Hath. 32 (1970), E. K. va Douwe, A techique for costructig hoest locally compact submetrizable spaces, preprit. 12. E. K. va Douwe ad H. H. Wicke, A real~ weird topology o the reals, preprit. 13. J. Ger1its, O Go p-spaces, Colloq. Math. Soc. Jaos Bo1yai 8 (1972), R. F. Gittigs, Some results o weak coverig coditios, Caad. J. Math. 26 (1974), , Cocerig quasi-complete spaces, Ge. Topology App1. 6 (1976), , Products of geeralized metric spaces, preprit. 17. R. E. Hodel, Moore spaces ad w~-spaces, Pacific J. Math. 38 (1971), T. Ishii, O wm-spaces. I~ II. Proc. Japa Acad. 46 (1970), R. L. Moore, A set of axioms for plae aalysis situs, Fud. Math. 25 (1935), K. Morita, Products of ormal spaces with metric spaces, Math. A. 154 (1964), Y. Taaka, O ope fiite-to-oe maps, Bull. Tokyo Gakugei Uiv., Sere IV, 25 (1973), J. M. Worrell, Jr. ad H. H. Wicke, Characterizatios of developable topological spaces, Caad. J. Math. 17 (1965), Brookly College of the City Uiversity of New York Brookly, New York 11210