PRESERVATION OF CERTAIN BASE AXIOMS UNDER A PERFECT MAPPING

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1 Volue 1, 1976 Pages PRESERVATION OF CERTAIN BASE AXIOMS UNDER A PERFECT MAPPING by Deis K. Bure Topology Proceedigs Web: Mail: Topology Proceedigs Departet of Matheatics & Statistics Aubur Uiversity, Alabaa 36849, USA E-ail: topolog@aubur.edu ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volue PRESERVATION OF CERTAIN BASE AXIOMS UNDER A PERFECT MAPPING Deis K. Bure 1. Itroductio Suppose X ad Yare topological spaces ad f: X ~ Y is a perfect appig (i.e., f is closed, cotiuous, oto, ad f-l(y) is copact for every y E Y). There are several theores i the literature which idicate that certai base axios are preserved uder such a ap. Two iportat results of this type were give by Worrell ad Filippov: Theore 1.1 [Wo]: If X is developable ad f: X ~ Y is a perfect appig the Y is developable. Theore 1.2 [Fi]: If a T space X has a poit-coutable l base ad f: X ~ Y is a perfect appig the Y has a poitcoutable base. A alterate approach to the proof of Theore 1~2 was give by the followig characterizatio of spaces with a poit-coutable base. Theore 1.3 [BM]: The followig properties of a space Y are equivalet: (a) Y has a poit-coutable base. (b) Y has a poit-coutable cover 9? such that if y E W with W ope i Y there is a fiite subcollectio ~r of ~i such that y E ( U j" ) 0 c ( U~)c Wad y E ~. If a T space X has a poit-coutable base ~ ad f: X ~ Y l is a perfect ap the the copact set f-l(y) itersects oly coutably ay ebers of ~[Mi] for every y E Y. If CJ! = f(ffi) = {f(b): B E ~B}, it follows easily that coditio (b) of

3 270 Bure Theore 1.3 is satisfied, so Theore 1.2 is a iediate corollary to Theore 1.3. Techiques siilar to those used i the proof of Theore 1.3 have bee used by the author to partially aswer the questio of whether the perfect iage of a quasi-developable space is quasi-developable. I the course of this ivestigatio a characterizatio of developable spaces was obtaied which gives Worrell's result (Theore l.l) as a corollary. This characterizatio is give below, as well as the partial results for quasidevelopable spaces. We coclude the paper by icludig a proof of the result that the perfect iage of a space with a a-poitfiite base has a a-poit-fiite base [Fi], ad by givig a exaple to show that the correspodig result does ot hold for spaces with a a-disjoit base. 2. A Characterizatio of Developable Spaces I order to state ad prove the ai theore it will be ecessary to defie the idea of a pair-etwor ad develop soe copaio otatio. A collectio ':i = {(Qa' R ): a Ell.} of pairs of subsets of a a space X is called a pair-etwor for X if wheever x E W, with W ope i X, there is soe P = (Q,R ) E ~ such that x E Q c R a a a a c W. The otio of a pair-etwor is ot ew ad was used i [Ko] to defie a class of spaces which coicides with the class of sei-stratifiable spaces. If 9! is a pair-etwor for X ad P E ~ we let P' deote the first eleet i the pair P ad let P" deote the secod eleet. If it c '. let it' = {p': P E 9(,} ad it" = {P": P E 9(,}. I f x E X ad 9(, c: Sf let St(x, ffi.,) U{p ll : P E 9t, x E Pi}, ad if A c X the St(A,9t) U{p": P E 9t, A P' t- JiJ}. Whe 9l l,91,, it are subcollectios of ~ we defie 2

4 TOPOLOGY PROCEEDINGS Volue i gtl A ~2 A A 9t to be the collectio of all pairs of the for (Pi P 2 P~, Pi P 2... P~) such that Pi E 9t i, = 1,2,,. Recall that a quasi-developet [Be] for a space Y is a sequece { }~ of collectios of ope subsets of Y such that if x E U C Y where U is ope i Y, there is soe such that x E St(x, ) c U. Y is developable if ad oly if Y is quasi developable ad every ope subset of Y is a Fa-set [Be]. Theore 2.1. The followig properties of a space Yare equivalet: (a) Y is developable. (b) Y has a pair-etwor ~ = U se satisfyig.~ =l (i) Each 9!' is a loca l ly fii te co l lectio of closed sets ad 9?~ is a collectio of ope seth. (ii) Wheever C cue Y where C is copact ad U is ope~ there is soe E N such that C c St (c, se )cu. (c) Y has a pair-etwor 9t U ~ satisfyig,: =l (i) Each 9t, is a locally fiite collectio of closed se ts. (ii) Wheever x E U c Y with U ope~ there is soe E N such that x E (St(x,9t ))0 c U. Proof: (a) -+ (b). Let { }~ be a developet for Y where we ay assue +l refies. Sice Y is subparacopact [Bu] each has a closed refieet ~ U ~(,), where each =l ~f (,) is discrete. If = {G : a E A}' we ay assue each a ~ (,) ca be expressed as ~(,) {F(,o.): a E A} where F(,o.) C Go. for every a E A. Let ~(,) = {(F(,o.),G ): a E A}; o. the U{ i..i! (,):, E N} is a pair-etwor for X. :E'or ay fiite sequece,,, of positive iteqers, defie l 2 ~(l,2,,) = Cf(l, l ) " ~ (2,2) A '" Lf('~) Now suppose C cue Y where C is copact ad U is ope. Let

5 272 Bure x E C. Choose a sequece {i}:=l of positive itegers such that x is i soe eleet of ~(i,i) for every i E N. For each, let A U {Q': Q E ~(l,2,,)' Q' C t-~, Q". U}. Clearly {A}~ is a decreasig sequece of closed sets ad if each A is oepty there ust be soe z E ( A ) C. =l EN such that St (z, ) c: U; it follows that Let St (z, ~(l, 2,, )) c St (x, ) c:: U ad this will cotradict the defiitio of A. Thus A ~ for soe ad this iplies x c St(C, ~(l,,)) C U. Now let ~ = U {~(l,,):,,, is a fiite sequece of l 2 positive itegers}. Cosider all collectios obtaied by taig uios of a fiite uber of eleets of {~(l,,): l,, is a fiite sequece of positive itegers}. These collectios ca be euerated as Sf l, Sf 2, S 3' ad Sf, ca be expressed' as ~ = U ~ with ~ satisfyig the coditios give i (b). =l (b) ~ (c). Trivial. (c). (c) ~ (a). Let 9t U 9t be a pair-etwor as give i =l For every, E N let ad let = {1" c9t': 11"1 }., G(~) = (U{R": R E 9t ad R' E ~F })O - U (9t~ -~) for every ~E,- Defie, = {G(~): ~ E,}- Sice gt, is a a-locally fiite etwor of closed sets it is clear that ope subsets of Yare Fa sets ad it suffices to show that { :, EN} is a quasi-developet for Y. To show this, let x E U where U is ope; by assuptio there is soe EN s uch that x E (St (x, 9t) ) 0 cu. Let ~ f = {R': R E 9t ' x E R'}; the I ~f I > 0 for soe EN so ~f E <P,. Clearly x E G( ~f) c (St(x, 9t )) 0 c: U. If ~ E <P, such that 6J t- ~F the x E (9t~ - f9) ad x ~ G(~) _ This says that G( ~f) is the oly eleet of, which cotais x. Thus x E G(1) = St(x, ) c: U.,

6 TOPOLOGY PROCEEDINGS Volue That copletes the proof of the theore. To see that Theore 1.1 follows as a corollary to the precedig theore suppose X is developable ad f: X + Y is a perfect appig. Let ~ be a pair-etwor for X satisfyig the coditio as i (b) of Theore 2.1. If 9t = {(f(p I ), f(p It»: P E ~} it is easily verified that 9tis a pair-etwor for Y satisfyig coditio (c) of Theore 2.1, so Y is developable. 3. Quasi-developable Spaces We ow tur to the questio of whe a quasi-developet is preserved uder a perfect ap. I [BL] Beett ad Lutzer showed that if GlL is a ope cover of a quasi-developable space X the GlL has a refieet ~= U ~ such that each ~ is =l discrete relative to U {F: F E ~ } The ext lea exhibits a slightly stroger versio of this coverig property. Lea :3.1. Suppose {~}~ is a quasi-deve z,opet for a space X. If GlL is ay co Z, Z,ectio of ope subse ts of X there is a refieet 1" = U ~ of GlL such that each 1" is cz,osed ad =l discrete rez,ative to (U 6lL) (U ~). Proof: Assue 6lL = {U : a. E 1\} where 1\ is well-ordered. a. For each E N, a. E 1\, let p {x: x E U", - ( U U ), x E St(x, } c:: u } a,a. ~ S<a. ~ a. ad let F be the closure of P relative to (U~) U ( ~).,a.,a.. Let x E (U~) (U ) ad suppose a. is the first elleet of 1\ such that x E U. Clearly U P Q = ~ if S > a. so a. a.,~ U F = ~ if S > a.. If S < a. ad St (x, ~ ) F Q:I- ~, the a.,s,~ there is soe z E St(x, ) P a. This iplies,~ x E St(z' ) C US' a cotradictio to our choice of U. Hece a St(x, ) F a = ~ if S <a.. It follows that U St(x, ),~ a. is a ope set about x which has epty itersectio with F,S for ay S E 1\, S :I- a.. This says that ~ = {F a: S E 1\} is,~

7 274 Bure discrete relative to (U~) (U ) ad that FeU for 00,B B every BE A. If ~ = U ~ it is clear that U ~F = U~; that =l copletes the proof of the lea. Theore 3.2. Suppose f: X + Y is a perfect appig ad X has a quasi-developet { }~ such that wheever x E U f- l (y) where U is ope i X ad y E Y.the there is soe EN such that covers f-l(y) ad St(x, ) C U. The Y is quasi-developable. Proof: For each let H = U ad suppose {G : a E A}' We ay assue each H is saturated with respect a to f. By Lea 3.1, has a refieet U ~(,) where each =l ~(,) is closed ad discrete relative to H H ; we ay also assue ~(,) has the for ~(,) = {F : a E A } where each,a Let ~(,) = {(F ' G H H ): a EA };,a a the U {~(,):, E N} is a pair-etwor for X. For fiite sequeces,,, ad,,, of positive itegers, l 2 r l 2 r defie ~(l,l,2,2,,r,r) ~(l,l) A ~(2,2) A A ~(r,r) Now suppose f-l(y) cue X where y ~ Y ad U is ope. Let 1, 2, 3, be a sequece of positive itegers such that if ~ -1-1 ~ covers f (y) the = for soe i. Let x E f (y). i Choose a sequece {i}~=l of positive itegers such that x is i soe eleet of ~(.,.) 1 1 for every i. Usig a arguet siilar to that used i the proof of (a) + (b) i Theore 2.1 it follows that there is soe r E N such that -1 x E St(f (y), ~(l,l,2,2,,r,r))c U. Now let the faily of all collectios ~(l,l,,s,s)' where,,, ad,,, l 2 s l 2 s are fiite sequeces of positive itegers, be euerated as ~l' ~2' ~3' ' For a give ~., say ~. J J B. H H... H If MeN is a fiite set, J l 1 s

8 TOPOLOGY PROCEEDINGS Volue defie B = {B : j E M} ad let M j 9t = {(QI B, Q" B ): Q E ~j' j EM}. M M M The faily of all collectios fit where M is a fiite subset of M, 00 N, ca be euerated as se, ~, se, ad if ~ = U se the =l ~ satisfies: (1) Each ~ I is locally fiite ad closed relative to use". (ii) If f-l(y) c U where U is ope i X ad y E Y -1-1 there is soe EN such that f (y) c St(f (y), se ) C U. For every, E N let ci>, = {~ c ~~: I ~ I = } ad let G(~) be the saturated part (with respect of f) of (U{p": P E ~ pi E ~}) - U(se l -~). ' Defie (,) = {f(g(~»: ~ E ci> }; we show { (,):, EN}, is a quasi-developet for Y. Y. By (ii) above there is EN such that -1-1 (1) -1 f (y)cst(f (y),;r;.,)cf (V). Let y EVe Y where V i.s ope i Let ~= {pi: P E ~, f-l(y) pi :I ~}; the I ~I >O for soe iteger, so ~ E ci>. Clearly, f-l(y) C G(~) c St(f-l(y), se ) C f-l(v). l If 5, E ci> such that 6J :I ~ the f- (y) (U (se I - 6J» :I ~ ad, f-l(y) G(6J) =~. This says that f(g(~» is the oly eleet of (,) that cotais y. Thus y E f(g(j"» = St(y, (,» c V. That copletes the proof of the theore. I geeral, a give quasi-developet for a space X ay ot satisfy the hypothesis of Theore 3.2, however the quasi-developet ca ofte be odified i order to obtai the desired coditio. The ext corollary gives oe situatio i which this is always the case. A p-base (poit separatig ope cover) for a space X is a collectio ill of ope sets such that wheever x,y E X, x :I y, there is soe B E ill such that x E Bad y e B. Corollary 3.3. Let f: X + Y be a perfect ap. If X is

9 276 Bure quasi-developable ad has a coutable p-base the Y is quasidevelopable. Proof: Suppose { }~ is a quasi-developet for X ad ffi = {B : E N} is a coutable p-base. For every, E N let '1L, = {G B : G E }. Let X,X 2, be a eueratio I of all collectios obtaied by taig uios of a fiite uber of eleets of {GlL :, EN}. It is easily verified that if., x E C U where C c X is copact ad U is ope there is soe EN such that JC covers C ad St(x,X ) c U. The corollary ow follows fro Theore 3.2. Corollary 3.4. Suppose X is Hausdorff ad f: X ~ Y is a perfect appig such that f-l(y) is a sigleto set for all but coutably ay y E Y. If X is quasi-developable the so is Y. Proof: Let E {y E Y: If- l (y) I > I}; the E is a coutable set. For each y E E the copact subspace f-l(y) of X is quasidevelopable ad thus separable etrizable [Be]. There is a coutable collectio ~(y) of closed subsets of f-l(y) such that wheever x,z E f-l(y), x ~ z, the there is soe F E ~(y) where x E F ad z ri:. F. Let 93= {X - F: F E ~(y), Y E E}; the ffi is a coutable ope cover of X such that wheever y E Y ad x,z E f-l(y), x ~ z, the there is soe BE ffi such that x E B, z e B. A costructio siilar to that used i Corollary 3.3 will ow fiish the proof. Corollary 3.4 ca also be prove directly without referece to Theore 3.2. I this case oe shows first that Y is first coutable ad the a quasi-developet for Y is costructed by cosiderig the poits of E separately. 4. Spaces With a d-poit Fiite Base A base ffi for the topology of a space X is said to be a poit-fiite if ffi ca be expressed as 93 = U ffi where each 93 =l

10 TOPOLOGY PROCEEDINGS Volue is poit-fiite. Filippov stated i [Fi] that the perfect iage of a space with a a-pait-fiite base has a a-pait-fiite base, but he did ot give a explicit proof. Soe recet iterest has bee show i seeig a proof of this result, ad sice a proof has ot appeared i prit we provide oe here. This proof was obtaied by the author several years ago while worig o soe related aterial with E. Michael. We begi with a lea that ay have soe idepedet iterest. Lea 4.1. If ~ is a poit-fiite collectio of subsets of X~ A c X~ ad E N~ the there are at ost a fiite uber of iial covers ~ of A~ by eleets of~~ such that I ~ I =. Proof: Suppose there is a ifiite collectio ~ of iial covers (of A) cosistig of subcollectios fro ~ of cardiality. Pic a axial collectio R.,c~ such that R.,c~ for ifiitely ay ebers ~ E ~, ad let ~I = {~E ~:fitc ~}. Cl,early o ~ I fit I <, so R., does ot cover A ad there is soe yea - (U R.,) Hece if ~ E ~I, there is soe F E ~ - fit such that y E F. Sice oly fiitely ay eleets of ~ cotai y, there ust be soe F E c;l such that y E F E ~ - O O fit for ifiitely ay ebers ~ E ~ I. The 9t U {F O} c ~ for ifiitely ay ebers ~ of ~, which cotradicts the axial coditio placed o 9t. Theore 4.2 [Fi]. If X has a a-pait-fiite base ad f: X ~ Y is a perfect appig the Y has a a-pait-fiite base. ffi Proof: Suppose ffi = U ffi is a ope base for X where each =l is poit-fiite. For each, E N, let ~ = { l' c U ffi.: I ~ I = }., i=l 1 For j"' E ~ 1et, <t ( 1') {A E U ffi.: l' is a iial cover of f- l (f (A» }. i=l 1 ad let U(~) = Y - f[x - U(~(1'»]. Defie GlL, = {U ( ~): J" E ~,} ad GlL = U {"1.L,:, E N}. To show that GlL is a base for Y, let yew c Y where W is ope i Y. Sice

11 278 Bure l f- (y) is copact, there is, E Nad:} E ~, such that 1" l is a iial cover of f- l (y) ad U 1" c f- (W). Now, if r > r adae U 93. suchthataf-l(y) ~~adf-l(f(a»c U~ i=l 1 the A E 'DlL (~); thus r ca be chose large eough so that r f-l(y) c U 'DlLr(~) ad it follows that y E U (1") c W. To cor plete the proof of the theore we show that each ~ is poit, -1 fiite. Let y E Y ad pic a fixed x E f (y). If -1 (;WJ Y E U ( ~) E G}L,_ (so ~ E ~ ) the f (y) c U ul"" (1") ad,1(", x E A for soe A E 'DlL (~); sice x E A for oly fiitely ay A E U 93. it suffices 'to prove that each A out of U 93. is i i=l 1 i=l 1 oly fiitely ay 'DlL (:}) for ~ E ~,. But this follows fro Lea 4.1 ad the defiitio of the 'DlL ( :}). That copletes the proof of the theore. The followig exaple, due to R. W. Heath ad G. M. Reed, shows that a a-disjoit base is ot ecessarily preserved uder a perfect appig. ExapZe 4.3. There is a exaple of a Moore space X with a a-disjoit base ad a perfect appig f: X ~ Y where Y does ot have a a-disjoit base. If R is the set of real ubers let H = {(x,y) E R x R: y>o}, X o = R x {O}, Xl = R x {-I}, ad X = H U X o U Xl. Descr~be a local base for each poit as follows: All poits i Hare isolated i X. If a E Rad E N, let U (a,o) = {(a,o)} U {(x,y) E H: x = y + a, y < li} ad U (a,-l) = {(a,-l)} U {(x,y) E H: x = -y + a, y < li}. The {U(a'O)}~=l ad {U(a,-l)}~=l are local bases at (a,o) ad (a,-l) respectively. It is easily verified that this iduces a topology o X aig X a regular, developable space with a a-disjoit base. Let Y be the quotiet space obtaied fro X by idetifyig the poits (a,o) ad (a,-i) for each a E R, ad

12 TOPOLOGY PROCEEDINGS Volue let F: X ~ Y be the correspodig quotiet ap. The f is a perfect ap, ad Y does ot have a a-disjoit base. This last fact ca be show directly, or it ca be oted that Y is hoeoorphic to the space described i Exaple 1 of [He]. Heath has show this exaple is a oscreeable Moore space, ad hece could ot have a a-disjoit base. Refereces Be BL BM Bu H. R. Beett, O quasi-developable spaces, Ge. Top. Appl. 1 (1971), H. R. Beett ad D. Lutzer, A ote o wea 8-refiability, Ge. Top. Appl. 2 (1972), D. Bure ad E. Michael, O a theore of v. v. Filippov, Israel J. Math 11 (1972), D. Bure, O subparacopact spaces, Proc. Aer. Math. Soc. 23 (1969), Fi V. V. Filippov, Preservatio of the order of a base uder He Ko Mi Wo a perfect appig, Soviet Math. Dol. 9 (1968), R. Heath, Screeability~ poitwise paracopactess~ ad etrizatio of Moore spaces, Caad. J. Math. 16 (1964), Ja. Kofer, O a ew class of spaces ad soe probles of syetrizability theory, Soviet Math. Dol. 10 (1969), A. S. Misceo, Spaces with a poitwise deuerable basis, Soviet Math. Do1. 3 (1962), J. Worrell, Upper sei-cotiuous decopositios of developable spaces, Proc. Aer. Math. Soc. 16 (1965), r.1iai Uiversity Oxford, Ohio 45056