ON THE M 3 M 1 QUESTION

Transcription

1 Vlume 5, 1980 Pages ON THE M 3 M 1 QUESTION by Gary Gruehage Tplgy Prceedigs Web: Mail: Tplgy Prceedigs Departmet f Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA tplg@aubur.edu ISSN: COPYRIGHT c by Tplgy Prceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Vlume ON THE M3 ~ Ml QUESTION Gary Gruehage 1. Itrducti I 1961, J. Ceder [C] defied the Mi-spaces, i = 1,2,3, prved that M ~ M ~ M ad asked whether ay f the l 2 3, implicatis reversed. H. Juila [J] ad the authr [G ] l idepedetly prved that M 3 -spaces, usually called "stratifiable spaces," are M 2. The questi remais whether M ~ M that is, whether every stratifiable space has a 3 l, a-clsure-preservig base. Sme partial results btaied s far are that the clsed image f a metric space is M l (F. Slaughter [Sl]), ad that a-discrete stratifiable spaces are M [G ]. Recetly, R. Heath ad Juila shwed that l 2 every stratifiable space is the image f a Ml-space uder a perfect retracti (ad hece is a clsed subset f a Ml-space) Let us call a space which is a cutable ui f clsed metrizable subspaces a Fa-metrizable space. I the first part f this paper, we prve that every stratifiable Fa-metrizable space is MI. May cmm examples f stratifiable spaces seem t be f this type. Fr example, all the examples give i Ceder's paper, as well as the chuk-cmplexes (which he prves t be M I ), are Fa-metrizable. Hyma's M-spaces [H], called paracmplexes ad prved M I by J. Nagata [N], are als f this type. Ather iterestig class f stratifiable spaces is the fllwig. Let I be a idex set, ad fr each i E I,

3 78 Gruehage let Xi be a stratifiable space. Let p E D X., where "0" iei 1 detes the bx prduct. Let y {x E 0 X.: x{i) = p{i) iei 1 fr all but fiitely may i E I}. Brges [B ] prved that 3 if each Xi is stratifiable, s is Y. It is t hard t shw that if each Xi is Fa-metrizable, the s is Y; hece Y is M l i this case. I [G ], we asked whether a stratifiable space which 2 has a a-discrete etwrk csistig f cmpact-sets is M l. Sice cmpact stratifiable spaces are metrizable, ur result implies a affirmative aswer t this questi. Ufrtuately, the class f stratifiable Fa-metrizable spaces is t clsed uder clsed maps. I fact, the clsed image f a metric space eed t be Fa-metrizable [F]. It must be M thugh, by Slaughter's result mel, tied abve. Nw suppse a space X has the fllwig prperty: wheever Had K are clsed subsets f X with H c K, the H has a a-clsure-preservig uter base i K (i.e., there is a a-clsure-preservig cllecti lj f relatively pe subsets f K such that wheever H c 0, 0 pe, there exists U E lj with H cue 0). I the secd secti f this paper, we prve that if a Ml-space X has the abve prperty, the every clsed image f X is M ad l has the same prperty. Frm the fact that stratifiable Fa-metrizable spaces are M it is easily shw that they l, als have the abve prperty. Thus every clsed image f a stratifiable Fa-metrizable space is M This geeralizes l Slaughter's therem, ad aswers a questi f Nagata ccerig the paracmplexes metied abve.

4 TOPOLOGY PROCEEDINGS Vlume Nagata als shwed that if X is a paracmplex, the Id X < if ad ly if X has a a-cls~re-preservig base B such that Id(aB) < - 1 fr every B E B. He asked if this result is true fr ay Ml-space. Mizkami [M] shwed that it is true fr a Ml-space which is Fa-metrizable ad satisfies a certai further cditi. With ur techiques, we ca shw it is true fr ay stratifiable Fa-metrizable space. 2. Defiitis ad Other Prelimiaries All spaces are assumed t be regular. Let A O dete the iterir f a set A. A cllecti ~ f subsets f a space X is iterir-preservig if wheever ~' c ~, the (~') = {G : G E ~'}. A cllecti H f subsets f X is alsure-preservig if wheever H' c U{H: H E H'}. H, the uh' It is easy t see that the set f cmplemets f a iterir-preservig family is clsure-preservig, ad vice-versa. If H is a subset f a space X, a uter base fr H is a cllecti lj f pe subsets f X such that wheever H is ctaied i a pe set 0, the there exists U E lj such that H c U c:. A cllecti B is a quasi-base fr X if wheever x E U, U pe, there exists B E B such that x E B O c B c U. (B detes the iterir f B.) A space X is a Ml-spaae (M 2 -spaae) if X has a a-clsure-preservig base (quasibase). A M 3 -spaae, r stratifiable spaae, is the same as a M 2 -space.

5 80 Gruehage We will fte use the fllwig characterizati f M 2 -spaces due t Nagata [N ]. 2 Therem 2.1 (Nagata). A regular space X is a M 2 -space if ad ly if fr each x E X ad E W 3 there exists a pe eighbrhd g(x) f x such that (1) y E g (x) :::;> g (y) c g (x); ad (2) if H is clsed ad x ~ H 3 there exists Ew such that x f U 9 (x). xeh Clearly, we may assume g(x) ~ gl(x) A cllecti J is a etwrk fr X if wheever x E U, U pe, there exists F E J such that x E Feu. X is a a-space if X has a a-discrete etwrk. A space X is mtically rmal if fr every pair (H,K) f disjit clsed subsets, there exists a pe set D(H,K) such that (1) H c D(H,K) c D(H,K) c X - K; (2) H c H' ad K ~ K' => D(H,K) c D(H',K'). We shall be usig the fact that stratifiable spaces are paracmpact ad perfectly rmal [e], that they are a-spaces [H], ad that they are mtically rmal [HLZ]. Als, every subspace f a stratifiable space is stratifiable [C], ad every clsed image f a stratifiable space is stratifiable [B]. 3. Mai Results, Outlies fthe Prfs, ad Sme Questis Sice the prf f ur mai results are rather lg ad tedius, we will defer them t later sectis, givig ly brief utlies here.

6 TOPOLOGY PROCEEDINGS Vlume Therem 3.1. Let X be stratifiable ad Fa-metrizable. The X is MI. Als~ Id X ~ if ad ly if X h~r a a-clsure-preservig base B such that Id(aB) < -l fr each B E B. Outlie f prf. Suppse X is stratifiable. The there exist g(x) 's satisfyig the cditis f Therem 2.1. The stadard way t get a a-clsure-preservig quasi-base back frm the g(x) 's is as fllws. Fr each clsed set H, defie G(H) = U g(x). xeh frm prperty (1) f Therem 2.1 that Y = {G(H): H is clsed, Hex} is iterir-preservig. Hece, B = {X - G (H): H clsed, Hex} is clsure-preservig, ad frm prperty (2), clsed quasi-base. preservig base wuld be t defie It is easy t see U B is a Ew A aive attempt t get a a-clsure f B = {(X - G (H)): H clsed, HeX}. Hwever, B' may fail t be clsure-preservig. But it turs ut that if the G(H) 's are regular pe sets, the B' will be clsure-preservig. (See Lemma 5.1.) S we cstruct g(x) 's satisfyig the cditis f Therem 2.1 s that the crrespdig G(H) 's are regular pe. We d this by cstructig a certai sequece VO,V l, f lcally fiite pe cvers f X, ad the use the Vi's t cstruct the g(x) IS, s that: (i) each g(x) is a elemet f sme Vi; (ii) fr each m E w, every ui f elemets f U V.. 1 l<m is regular pe; ad

7 82 Gruehage (iii) if H is clsed ad y f u g (x), the there exists XEH a iteger k such that y f Cl (U{g(x): x E H, g (x) f u [I.}). i<k 1 It easily fllws that G (H) = U g (x) is regular pe xeh wheever H is clsed. The Fa-metrizable hypthesis is used t btai prperty (iii). It is pssible t cstruct [10,[11' ad the g(x)'s satisfyig (i) ad (ii) i ay stratifiable space. The "if" part f the last statemet f Therem 3.1 is a result f Nagata [N ]. T btai the "ly if" part, 2 we shw that if Id X <, we ca cstruct the V.'s s 1 that Id(aV) < -l fr each V E Vi. It the fllws that Id(aB) < -l fr each B E B' Therem 3.2. Suppse X is stratifiable ad has the fllwig prperty: wheever H ad~k are clsed subsets f X with H c K, the H has a a-clsure-preservig uter base i K. The every clsed image f X has the same prperty, ad is therefre M I Outlie f prf. Let (*) dete the prperty f Therem 3.2. Ay stratifiable space satisfyig (*) is MI. This fllws easily frm the facts that every clsed subset has a a-clsure-preservig uter base, ad that stratifiable spaces are a-spaces. Let f be a clsed map f X t Y, where X is stratifiable ad satisfies (*). Sice every clsed subset K f Y is the clsed image f a stratifiable space satisfyig (*), amely f-l(k), it is eugh t shw that every clsed

8 TOPOLOGY PROCEEDINGS Vlume image f a stratifiable space satisfyig (*) has the prperty that every clsed subset has a a-clsure-preservig uter base. S we are de if we shw that Y has this prperty. li -1 By a therem f Okuyama [0], Y = Y' U y, where f (y) is cmpact fr each y E Y', ad y" is a-discrete. We use this t shw that Y Y U Y where Y is a clsed l, irreducible image f a clsed subset X f X, ad Y is l pe ad a-discrete. Frm results i [BL], it fllws that every clsed subset f Y has a a-clsure-preservig uter base i Y. Thus Y ca be writte as the ui f a clsed subspace havig the prperty we wat, ad a pe a-discrete subspace. The fial step is t shw that ay stratifiable space which admits such a decmpsiti als has the prperty that every clsed subset has a a-clsure-preservig uter base. Remark. Fr stratifiable spaces, prperty (*) is equivalet t the fllwig prperty: wheever Had K are clsed subsets f X with H c K, the K/H is MI. Crllary 3.3. The clsed image f a stratifiable Fa-metrizable space is MI. Prf. Suppse X is stratifiable ad Fa-metrizable. Let H eke X, where Had K are clsed. stratifiable ad Fa-metrizable, hece MI. The K/H is By the abve remark, X satisfies the cditis f Therem 3.2. It is t kw whether every M1-space satisfies the prperty f Therem 3.2. I fact, it is t kw if

9 84 Gruehage every clsed subset f a Ml-space is MI. Hwever, the result f Heath ad Juila metied i the itrducti implies, as they te, that this questi is equivalet t the M ~ M questi. It is als t kw whether every 3 I clsed subset f a Ml-space has a a-clsure-preservig uter base. But if t, the by results f Ceder, there is a stratifiable space which is t MI. O the ther had, Brges ad Lutzer [BL] have shw that if each pit f a stratifiable space has a a-clsure-preservig base, the every stratifiable space is MI. Brges ad Lutzer have als shw that if every clsed subspace f a space X is M the every perfect image f I, X is MI. This suggests the fllwig questi, which wuld geeralize Therem 3.2 if aswered affirmatively. Questi 3.4. If every clsed subspace f a space X is M is every clsed image f X als M? I, I A class f spaces which Heath ad Juila called M-spaces may have a imprtat rle t play i settlig the M ~ M 1 questi. A M-space is a space which has 3 a a-clsure-preservig base f pe ad clsed sets. It is easy t see that every subspace f a M-space is M. Thus every perfect image f a M-space is MI. Recetly, Juila [J ] has btaied a alterate prf that strati 2 fiable Fa-metrizable spaces are M I are perfect images f M-spaces. by shwig that they Heath ad Juila ask whether every stratifiable space is the perfect image f a M-space. If s, the M => MI. Our ext crllary 3

10 TOPOLOGY PROCEEDINGS Vlume shws that it wuld be eugh (fr the purpse f btaiig M =>M ) t prve that every stratifiable space is the 3 l clsed image f a M-space. Crllary 3.5. The clsed image f a M-space is Prf. We shw that every M-space X satisfies the prperty f Therem 3.2. If K c X, the K is M. By mimicig Ceder's prf that every clsed subset f a M -space has a clsure-preservig uter quasi-base, we see 2 that every clsed subset f K has a clsure-preservig base i K. The class f stratifiable Fa-metrizable spaces is t clsed uder clsed maps r cutable prducts. These are still M f curse. I fact, sice prducts f perfect l, maps are perfect, the cutable prduct X f stratifiable Fa-metrizable spaces is the perfect image f a M-space. Hece X satisfies the prperty f Therem 3.2, ad s every clsed subspace ad clsed image f X is MI. But further iteratis f the prcedures f takig clsed subspaces, clsed images, ad cutable prducts, prduces spaces that I ca't prve are MI. What e might aim fr is a sluti t the fllwig: Prblem 3.6. Fid a class f Ml-spaces which ctais the Fa-metrizable spaces, ad which is clsed uder clsed subspaces, clsed images, ad cutable prducts. Nte that the class f stratifiable F -metrizable a spaces is clsed uder arbitrary subspaces, perfect images,

11 86 Gruehage ad fiite prducts. The class f stratifiable spaces satisfyig the prperty f Therem 3.2 is clsed uder clsed subspaces ad clsed images. Thus e wuld have a sluti t the Prblem 3.6 if e culd shw that this prperty is clsed uder cutable prducts. Fr ather apprach, te that the class f perfect images f M-spaces satisfies all the desired prperties except perhaps clsure uder clsed images. Althugh we d't d it here, the techiques f secti 6 ca be used t shw that if a stratifiable space X is the ui f a clsed Ml-space ad a Fa-metrizable space, the X is MI. This suggests a cuple f questis, fr which affirmative aswers t bth (r a egative aswer t e) wuld bviusly settle the M ~ M questi., 3 I Questi 3.7. If a stratifiable space X is a cutable ui f a clsed M subspaces, is X M? I I Questi 3.8. Is every stratifiable space the cutable ui f clsed M 1 subspaces? 4. Prelimiary Lemmas I this secti we preset a series f lemmas regular pe sets, leadig up t the result that i a paracmpact hereditarily rmal space, every pe cver has a lcally fiite refiemet V such that every ui f elemets f V is regular pe. I fact, if W is ay lcally fiite cllecti such that every ui f elemets f W is regular pe, the V ca be cstructed s that

12 TOPOLOGY PROCEEDINGS Vlume every ui f elemets f V U W is regular pe. Lemma 4.1. Let X be a hepeditapily pmal space. Suppse U~ V~ ad U U V ape pegulap pe~ ad H is a pelatively clsed subset f v. Suppse H is ctaied i a pe set. The thepe is a set W such that HeW c V~ wc O~ ad bth Wad U U W ape pegulap pe. If X is pepfectly pmal~ Id X < ~ ad Id(aU) < -l~ the we ca btai Id(aW) ~ -l. Ppf. Let 0' be a pe set such that H c 0' c a' c O. Usig the hereditary rmality X, we ca fid pe sets VI ad V such that 2 -v -v H c VI c VI c V 2 c V 2 c V 0', where XV detes the clsure f A i the subspace V. 0 let W = ~V-I--U-U~ v We claim that W has the desired 2. prperties. Nw Clearly, Hew c V, ad Wc O. Als, sice the itersecti f tw regular pe sets is regular pe, W is regular pe. It remais t shw W U U is regular pe. T see this, suppse p E W U U - W U U. Observe that w-u-u c VI U U. Thus p E V--I--U-U- - U. Hece p E VI' sice U is regular pe. Als, p E v-u-u - U (V U U) - U, s P E V. Hece p E VI V c V 2. S we have 0 p E =V-l~U-u~ V = W, ctradicti. 2 U au. T see the last statemet, te that aw c av U av l 2 Sice Id V <, we ca btai Id(aVV i ) ~ -l, i = 1,2, where "a " detes the budary f a set i the v subspace v. Nw av = [av av] U avv Thus Id(aV ) i i i. i ~ -l, ad s Id(aV U av U au) < -l. Thus Id(aW) < -l. l 2

13 88 Gruehage Lemma 4.2. If Uis a cllecti f subsets f a space X such that every fiite ui f elemets f Uis regular pe, the every fiite ui f fiite itersectis f elemets f U is regular pe. Prf. This fllws frm the fact that a fiite ui f fiite itersectis f elemets f Uca be writte as the fiite itersecti f fiite uis f elemets f U, ad that a fiite itersecti f regular pe sets is regular pe. Lemma 4.3. Suppse Uis a fiite cllecti f pe sets f a hereditarily rmal space X such that every ui f elemets f Uis regular pe. Let H C 0, H clsed, pe. The there exists a set W such that HeW c 0, ad every ui f elemets f U U {W} is regular pe. If X is perfectly rmal, Id X ~, ad Id(aU) < -l fr each U E U, the we ca btai Id(aW) < -l. Prf. If I U/ = 1, apply Lemma 1 with U the elemet f U, V = X, ad Had 0 as i the hypthesis f this lemma. The set W guarateed by Lemma 4.1 is easily see t satisfy the desired cditis. Nw suppse lui = > 1. Let U = {U,ul,---,U l}. - Let W be a regular pe set such that HeW ewe 0, 0 ad (uu) u W is regular pe. Suppse W has bee defied, k <. Fr each subset k I f k + 1 distict elemets f {O,l,---,-l}, let H(I) (aw ) ( U.). The the hyptheses f Lemma 4.1 k iei 1 are satisfied with U = U U., V U., H = H(I), ad 0 jfi J iei 1 as i this lemma. (Nte the use f Lemma 4.2 t get U U V

14 TOPOLOGY PROCEEDINGS Vlume regular pe.) Let W(I) be the set give by Lemma 4.1, ad let W = W U (U{W(I): le, III = k + I}). k + l k Let W = W. We claim that W has the desired prperty. T see this, assume mc U U {W}, ad x E um - Um. Clearly, we may assume W E mad x E W. Let I = {i < : x E Ui}' ad let m = I II If m = 0, 0 0 the x E (UO) U W - ( UU) U W = (uo) u W - (UU) U W 0 0' ctradicti. S we may assume m > 1. The x f Wm-l' fr therwise x E aw m _ l ( U.) = H (I) c W(I) c W. S iei 1 there is a pe set G such that x E G c (Um) ( U.) (X - W 1). iei 1 m- Sice W(I) U ( U U.) is regular pe ad des't cjfi J tai x, there exists y E G with Y f W(I) U ( u U.). jfi J Y E um, s Y E U(m - {W} ) r yew. Sice U(m - {W}) c U U., it must be true that yew. Thus there exists a jfi J empty set J c such that y E W(J). Sice y f W l, m But we have IJI > m. Sice y f W(I), we have J ~ I. Therefre, there exists j E J - I. But W(J) c --u. c u., iej 1 J ctradictig y f -u-u.. jfi J Thus W has the desired prperty. The last statemet f Lemma 4.3 fllws because W is the ui f a fiite umber f sets btaied frm the applicati f Lemma 4.1. Lemma 4.4. Let U be a pe cver f a paracmpact hereditarily rmal space X, ad let V be a lcally fiite cllecti such that every ui f elemets f V

15 90 Gruehage is regular pe. The there is a lcally fiite refiemet W f lj such that every ui f elemets f V U W is regular pe. If X is perfectly rmal, Id X ~, ad Id(aV) < -l fr each V E V, the we ca btai Id(aW) < -l fr each W E W. Prf. Let J U I be a lcally fiite pe cver Ew f X by sets whse clsures refie lj ad meet ly fiitely may elemets f V, ad such that each J is discrete. Let ~ = {G F : F E J} be a shrikig f J; i.e., ~ is a pe cver such that G F c F fr each F E J. Let V(F) = {V E V: V F ~ ~}. Fr each F E J ' let W be the set F give by Lemma 4.3 where lj = V(F), H = G ad F. F, The G c W c F, ad every ui f elemets f V U {W F F F : F E J } is regular pe. Let W Nw suppse that fr each F E J k, k <, we have defied a W such that F (i) W = {W : F E J is a discrete cllecti f k F k } pe sets; (ii) if F E J k, the G - U (uwj ) c W c F; F F j<k (iii) if F E J k, the W F G, F = ~ wheever F' E J. J with j < k; (iv) every ui f elemets f V U ( u W) is regular j j~k pe. TO defie W fr F E F I, let 0 = {OF: F E I } be a discrete cllectis f pe sets such that G F - u (uw k ) c OF c F, ad k< OF (U {G F : F E J k' k < }) ~

16 TOPOLOGY PROCEEDINGS Vlume This is pssible, sice U{G : F E J k < } c U (UW ). F k, k< k Nw, fr each F E I, let W be the set give by Lemma 4.3 F applied t the case where U= V(F) U {uw k : k < }, H = G F u (uw k ), ad 0 = OF. k< It ia easy t check that W = {W : F E J} satisfies F prperties (i)-(iii) abve. T see that (iv) is satisfied, suppse x E um - Um, where V U ( u W k ). Sice this k< cllecti is lcally fiite, we may assume mis fiite ad x E M - M fr every M E m. Let I = {k < : m W ~ ~}. k Fr each i E I, x ~ fr sme W E W. 1 uw. ]. sice W. 1 is discrete ad x E W- W By the iducti hypthesis, we ca assume there exists W E m W, where F E J. The F m V = m V(F), sice x E W c F. Thus x is i the F iterir f the clsure f but is t i this set, ctradictig the way W was defied F (i.e., the abve set must be regular pe). fies (iv). Thus W satis Let W= U W That Wis a refiemet f U fllws Ew frm prperty (ii). That Wis lcally fiite fllws frm (i) ad (iii). Ad that every ui f elemets f V U W is regular pe fllws frm (iv) ad the lcal fiiteess. Let us see hw t btai the last statemet f Lemma 4.4. First te that uder the hyptheses we ca get Id(aW) ~ -l fr W E W ' sice these sets were als btaied frm Lemma 4.3. If it's true fr W E U W the k, k< it's als true fr W E W, sice these sets were als

17 92 Gruehage btaied frm Lemma 4.3, with the U f Lemma 4.3 beig a subset f V tgether with {UW : k < }, ad Id(d(UW )) k k < -l fr k < sice each W is discrete. k 5. Stratifiable F-Metrizable Spaces I this secti, we prve that stratifiable Fa-metrizable spaces are MI. First, a easy lemma. Lemma 5.1. Let U be a iterir-preservig cllecti f regular pe subsets f a space X. The {(X - U): U E U} is clsure-preservig. Prf. Suppse U' c U, ad x E U{ (X - U): U E U/}. Suppse fr each U E U', we have x f (X - U)O. Sice each U is regular pe, we must have x E U fr each U E U'. Thus x E U', which is a pe set missig (X - U)O fr every U E U'. This ctradicti establishes the lemma. Prf f Therem 3.1. Let X be a stratifiable Fa-metrizable space. Let X U M, where each M is a Ew clsed, metrizable subspace f X. Let M, mew,m where 0 is pe.,m Let {U'} be a sequece f relatively pe cvers,m mew f M which is a develpmet fr M Fr each U' E U',m' let U be pe i X such that U M U', ad U c 0,m Let U = {U: U' E U' },m,m Fr each x E X ad Ew, let g(x) be a pe eighbrhd f x such that the g(x) 's satisfy the cditis f Therem 2.1, ad that g(x) = X fr each x E X. Let J = u I be a clsed etwrk fr X such that each I is Ew

18 TOPOLOGY PROCEEDINGS Vlume discrete. Fr each F E I, let U F be a pe set such c ~ wheever F ' E J, F ' ~ F. Let V U ({gi(x): i < ad F c gi(x)}). Sice the F F gi (x) I s satisfy prperty- (1) f Therem 2.1, V is pe. F Nw use Lemma 4.4 t iductively cstruct a sequ~ce V,V, 6f lcally fiit.e pe cvers f X such that 1 (i) V + star-refies V ; l (ii) V refies {V : F E J.} u {X - uj.} fr each i < ; F 1 1 (iii) V refies U.. u {X - M.} fr each i,j _< ; 1,J 1 (iv) every ui f elemets f U V. is regular pe.. 1 l< It is easy t see frm Lemma 4.2 that we may assume that if x E X, the {V E V.: x E V, i < } is a elemet f V 1 Claim I. Fr each x E X, fr each Ew, there exists mew such that st(x,v ) c g (x). m Prf f Claim I. There exists l > ad F E J I such that x E F l c g(x). (T get ~, we ca assume that each I is repeated ifiitely fte.) If x EVE V l, the by (ii), V c V = U ({gi (y): i 2. l ad F c gi (y)}) F F g (x). l Let m = + 1. The st(x,v ) is ctaied i sme m elemet f V I' s st (x, V ) c m g (x) Claim II. If H is clsed i x, ad y f H M, the there exists mew ad V E V such that y E V ad st(v,v ) m m H M ~. Prf f Claim II. If Y f M, there exists l such that y ~ 0 I. Let m = + l + 1, ad pick V E V with, m y E V. The st (V, V ) ewe V + W must be ctaied i I. m

19 94 Gruehage sme eleiet f U,U {X Each elemet f U,is,, ctaied i 0," s W c X - M. Thus st(v,v ) H m M ~. If Y E M, there exists " E 00 such that st(y,lj ), H M =~. Let m = + + 1, ad pick V E V with m y E V. The st(v,v ) ewe V+. But W c U fr sme m U E U ", where y E U M E lj' ". Thus U H M ~,,, s st(v,v ) H M ~. m Fr each x E X, let j(x) E 00 be such that x E Mj(x) U M.. Frm Claims I ad II, it is easy t see that, i<j (x) 1 fr each Ew, there is a least iteger l(,x) such that st(x,vl(,x» c g(x) - U M.. We defie g~(x) = {v E Vi: i<j (x) 1 x EVE V., i < l(j(x) +, x)}. By the statemet immedi 1 ately precedig Claim I we have g~(x) E Vl(j(x)+,x). It's t re~lly ecessary t have this, but we said we culd get it i the utlie i secti 3. Claim III. If Y E g~(x), the g~(y) c g~(x). Prf f Claim III. If Y E g~(x), the y E gj(x)+(x) c g(x), s g(y) c g(x). Sice g~(x) (U M.) = ~, i<j (x) 1 we have j(y) ~ j(x). Thus gj. (y)+(y) - U M. c i<j (y) 1 gj.(x)+(x) - U M. If l(j(y)+,y) < l(j(x)+,x), the i <j (x) 1 we wuld have st(x,vl(j(y)+,y» c st(y,vl(j(y)+,y» c gj. (y)+(y) -.U. M i c gj. (x)+(x) - U M.. This cl<j (y) i<j (x) 1 tradicts the defiiti f l(j(x)+,x). Thus l(j(y)+,y) > 1 (j (x) +,x), ad s 9 I (y)/ c 9 I (x) - I

20 TOPOLOGY PROCEEDINGS Vlume Fr each clsed set H, defie G(H) = U g' (x). By xeh Claim III, the cllecti {G (H): H clsed, HeX} is iterir-preservig. Sice g~(x) c g(x), the g~(x) 's satisfy prperty (2) f Therem 2.1, s if we defie B = {(X - G (H» 0: H clsed, Hex} the U B is a base fr X. By Lemma 5.1, each B is Ew clsure-preservig if G(H) is always regular pe. we are fiished after prvig the ext claim. S Claim IV. G(H) is regular pe. Prf f Claim IV. Suppse y E G(H) - G(H). There exists k E w such that y f u gk(x). xeh S, sice g~ (x) c gj (x) + (x), y $ Cl (U {g~ (x): x E H, j (x) > k}). By Claim II, fr each j < k, there exists m E wad j V. E V such that y E V. ad st(v.,v ) H M. = ~. J m. J J m j J J Let V = V. Suppse x E H, j(x) < k, ad g~(x) V ~ ~. j <k J Nw g~(x) is ctaied i sme W E Vl(j(x)+,x)' ad W Vj(x) ~~. If l(j(x)+,x) ~ mj(x)' the W c st(v. ( ) V ). But x E W H MJ.(x)' s J x, m j (x) x E st(v.( },V ) H M. (x)' ctradicti. Thus J x mj(x) J l(j (x)+,x) < m j (x). Let m = sup{m.}. j<k J Y ~ We see, the, that Cl (u{g~ (x): x E H, j (x) < k, 1 (j (x)+,x) ~ m}). Cmbiig this with the first paragraph, we have y t Cl(u{g~(x): x E H, l(j(x)+,x) ~ m}). Thus y is i the iterir f the clsure f thse g~(x) 's

21 96 Gruehage which are elemets f U V.. This c?tradicts the fact that i<m 1 uis f elemets f U V. are regular pe l ad the. 1 l<m prf that X is M I is fiished. Nw suppse Id X <. We will shw hw t btai Id(aB) < -l fr each B E U B. Sic~ a(a - Gk(H)) c Ew agk(h), we will be de if we ca get Id(aGk(H)) < -l fr a arbitrary k E wad clsed set H. By Lemma 4.4, we ca add the fllwig t the list f prperties f the sequece V,V I, : (v) Fr each Ew ad V E V, Id(aV) < -l. The sice each gk(x) is the itersecti f fiitely may members f U V., we have Id(agk(x)) < -l~ iew 1 Suppse y E agk(h). we see that there exists mew such that The by the prf f Claim IV, y t Cl (U {gk (x): x E Had gk (x) ~. U Vi}) l<m Thus there is a eighbrhd f y meetig ly fiitely may elemets f {gk(x): x E H}, ad s we have lc Id(aGk(H)) < -l. Sice X is hereditarily paracmpact, Id (dg (H» < -l. k 6. Clsed Images I this secti we prve Therem 3.2. We preset the mai part f the prf as a series f lemmas, sme f which may be f idepedet iterest. A map f: X ~ Y is irreducibze if prper clsed subset maps t Y.

22 TOPOLOGY PROCEEDINGS Vlume Lemma 6.1. Suppse X is stratifiable, ad f: X + Y is a clsed ctiuus surjecti. The there exists a clsed set X c X such that fl : X + f(x ) is irreducible, x ad Y - f(x ) is pe ad a-discrete. Prf By a therem f Okuyama [0], Y = Y U Y l, where each pit f Y has a cmpact pre-image, ad Y l a-discrete. Let C= {XI C X: Xl is clsed ad f(x I ) ~ Y}. is Partially rder Cby iclusi. It is easy t see frm the fact that f-l(y) is cmpact fr each y E Y that every chai [ i C has a lwer bud, amely [. Thus C has a miimal elemet X. X is clsed, ad Y - f(x ) C Y l, hece is a-discrete. The miimality f X fix: X + f(x ) is irreducible. [BL] implies that The ext lemma is essetially due t Brges ad Lutzer Lemma 6.2. If each clsed subset f X has a a-clsurepreservig uter base, ad f: X + Y is clsed ad irredu~ible, the each clsed subset f Y has a a-clsure-preservig uter base. Prf. Suppse Key is clsed. Let U = U U be a Ew is clsure-preserv uter base fr f-l(k) such that each 0 ig. Fr A C X, let f#(a) {y E Y: f-l(y) c A}. By [BL, Lemma 3.3], 0' = {f#(u): U EO} is clsure-preservig. Thus 0# = u 0# is a a-clsure-preservig uter base fr K. Ew Lemma 6.3. A clsed set K c X has a -clsure-preservig uter base if ad ly if fr each clsed HeX - K,

23 98 Gruehage there exists a sequece {G (H,K)} Ew such that (1) each G(H,K) is a regular pe set ctaiig H; (2) fr each E W 3 {G(H,K): H clsed 3 H K =~} is iterir-preservig; ad (3) fr each clsed HeX - K 3 there exists Ew such that G(H,K) K = ~. Prf. T see the "if" part, suppse we are give G (H,K)'s satisfyig (1)-(3). Let U = {(X - G (H,K»: G (H,K) K = ~}. By (3), U = U lj is a uter base fr Ew K. By (1), (2), ad Lemma 5.1, each U is clsure preservig. T see the "ly if" part, suppse U = U U is a Ew uter base fr K, where each U is clsure-preservig. Defie G (H,K) = X - U{U: U E U ad IT H = ~}. Clearly G (H,K) is pe ad ctais H. Sice U is a uter base fr K, (3) hlds. Sice the set f cmplemets f a clsure-preservig cllecti is iterir-preservig, (2) hlds. It remais t prve that G(H,K) is regular pe. Suppse x f G(H,K). The there is U E U with x E IT ad IT H =~. The U G(H,K) = ~, ad every pe set c taiig x meets U. Thus x f ~G--(H=-,K~) Lemma 6.4. Suppse Y = Y U Y 13 where Y is m tically rmal, Y is clsed 3 ad Y = Y - Y. Let K l be clsed i Y. If U is a iterir-preservig cllecti f relatively pe subsets f Y whse clsures miss K 3 the e ca assig t each U E U a set U* pe i Y such that

24 TOPOLOGY PROCEEDINGS Vlume U E U/}]. (1) U* Y u " (2) U* Y u; (3) U* K = ~; ad (4) if y E u'" where U' c U" the y E [iu*: Prf. Accrdig t [B 2, Therem 2.4], sice Y is mtically rmal, fr each x E Y ad pe eighbrhd U f x, e ca assig a pe eighbrhd U x that f x such (i) U c V => Ux c Vxi (ii) U x V y ~ l' => x E U r y E V. Nte that (ii) implies U c U. x Fr each U E U, let U* = U [(U U Y ) - K] That (1) xeu l x is satisfied is bvius. T see (2), suppse y E (U*' Y) - U. Let W be a pe eighbrhd f y such that W U ~~. Sice W U* ~ 1', there exists x E U such that y w [(U U Y ) - K)x ~~. This ctradicts prperty (ii) y l abve. T see (4), suppse y E U', where U' c U. The [ ( ( U ') U Y1) - K] y C [(U U Y1) - K] y c U* fr each U E UI. It remais t prve (3). Suppse y E U* K. Sice IT K = ij, it fllws frm (2) that y E Yl The (Yl)y U* ~ 1', s there exists x E U such that (Yl)y [U U Y ) l K]x ~ 1', agai ctradictig (ii) Lemma 6.5. Let X be stratifiable ad a-discrete" ad suppse fr each x E X we have assiged a eighbrhd O(x) f x. The e ca assig t each x E X a pe eighbrhd U(x) f x such that

25 100 Gruehage ( 1) U (x) c a (x) ; (2) y E U(x) ~ U(y) c U(x); (3) if HeX is ezsed 3 the U U(x) is pe ad xeh clsed. Prf. The prf is similar t that f [G Therem 2, 1]. Let X = U F, where each F is clsed discrete, ad Ew F F = ~ if m ~. Fr each x E X, let (x) be the m least iteger such that x E F(x). Let D be a mte rmality peratr fr X. Iductively, defie, fr each x E X, a set U(x) ctaiig x such that (i) {U(x): x E F } is a discrete cllecti f pe ad clsed sets; ad (ii) U(x) c a(x) D({x}, u F.) ({u(y): x E U(y) i < (x) 1 ad (y) < (x) }. Sice X is a-dimesial ad cllectiwise-rmal, the abve cstructi ca easily be carried ut. These U(x) 's clearly satisfy (1). Als, if Y E U(x), the (x) < (y), s U(y) c U(x) by (ii). Thus (2) hlds. Fially, t see (3), suppse H is clsed ad y f U U(x). xeh The D(H,{y}) ~ D({x}, u F.) fr all x E H with i<(x) 1 (x) > (y). Thus D (H, {y}) ~ U{U (x): x E H, (x) > (y) }, ad s y ~ U{U(x): x E H, (x) > (y)}. But U{U(x): x E H, (x) < (y)} is pe ad clsed. Thus y ~ U U(x). xeh Lemma 6.6. Suppse Y is stratifiable 3 Y = Y U Yl~ where Y is clsed, Y is a-discrete, ad Y Y l =~. If l every clsed subset f Y has a a-clsure-preservig uter

26 TOPOLOGY PROCEEDINGS Vlume base i Y' the every clsed subset f Y has a -clsurepreservig uter base. Prf. Fr a clsed set Hey, let H = H Y ad HI = H Y Let Key be clsed. We will shw that K has l. a -clsure-preservig uter base. Fr clsed Hey - K, let G(H,K ) be a relatively pe subset f Y havig the prperties f Lemma 6.3. If G(H,K ) K = ~; let G(H,K )* be as i Lemma 6.4. Otherwise, let G(H,K )* Y. Let {g(y): Y E Y, Ew} have the prperties f Therem 2.1. Let Y U' where each U is pe ad Ew ctais U + l. Fr each x E Y - K, let O(x) l = [(U(x) U (x)+2) g(x) (x)] - such that x E U(x). K, where (x) is the largest iteger Let U(x) be as i Lemma 6.5 applied t Y. 1 Nw fr H clsed, Hey - K, ad Ew, defie V (H,K) = G (H,K ) * U (U {U (x) : x E [G (H, K ) * U H] Y }). 0 0 l We will shw that prperties (1)-(3) f Lemma 6.3 hld fr the V(H,K) IS. First we will shw that V(H,K) Y = G(H,K ). Suppse t. The there exists Y E V(H,K) - G(H,K ). Observe by Lemma 6.5 that V(H,K) Y l is pe ad clsed i Y l. Thus Y E Y. By Lemma 6.4, y $ G(H,K )*. Als y $ H, s there exists Ew such that y ~ Cl(U{g (x): 0 x E G(H,K )* U H}). Thus y ~ C1(U{U(x): x E (G(H,K )* U H) Y U }). But C1(U{U(x): x E (G(H,K )* U H) l Yl (Y - U )}) c Y - U +1. Thus Y ~ V (H,K), ctradicti.

27 102 Gruehage Clearly V(H,K) is pe ad ctais H. Suppse y E V(H,K) - V(H,K). Sice V(H,K) Y l is clsed i Y l, we have y E Y V (H,K) = G (H,K ) Sice G (H.,K ) is regular pe i Y' each eighbrhd f Y ctais a pit Z E Y - G(H,K ) The z t V(H,K). Thus V(H,K) is regular pe, ad s prperty (1) f Lemma 6.3 hlds. T see (2), suppse II' is a cllecti f clsed sets missig K, ad y E V (H, K). We may assume V(H,K) ~ Y. HE II' The y t K. If Y E Y the U(y) C V(H;K) fr each H E l, II' If Y E Y' the ye G (H,K ), s by Lemma 6.4, HEll' 0 0 y E ( G (H,K )*)0 c V (H,K). Thus (2) hlds. HEll' 0 0 HEll' Fially, t see (3), let Hex - K be clsed. There ~. If Y E V(H,K) K, the sice V(H,K) Y G(H,K )' we have y E Y l But V(H,K) Y is clsed i Y ad misses K. Thus V(H,K) I I K = ~. Prf f Therem 3.2. Therem 3.2 prperty (*). Let us call the prperty f Suppse X is a stratifiable space satisfyig (*), ad let f: X ~ Y be a clsed map f X t Y. The Y is stratifiable. We eed t shw that Y satisfies (*). Let Key be clsed. Sice fl f-l(k) is clsed, by Lemma 6.1, there exists a clsed set K C K such that K is the clsed irreducible image f a clsed subset f f-l(k), ad K - K is a-discrete. By Lemma 6.2, ad the fact that every clsed subset f X satisfies (*), we see that every

28 TOPOLOGY PROCEEDINGS Vlume clsed subset f K has a a-clsure-preservig uter base i K. The by Lemma 6.6, every clsed subset f K has a a-clsure-preservig uter base i K. Thus Y satisfies (*) Refereces [B ] C. R. Brges, O stratifiable spaces, Pacific J. l Math. 17 (1966), [B 2 ], Fur geeralizatis f stratifiable [BL] [C] [F] spaces, Geeral Tplgy ad its Relatis t Mder Aalysis ad Algebra, Prague Sympsium, 1971, , Direct sums f stratifiable spaces, Fud. Math. C (1978), ad D. Lutzer, Characterizatis ad mappigs f Mi-spaces, Tplgy Cferece VPI, Spriger Verlag Lecture Ntes i Mathematics, N. 375, J. Ceder, Sme geeralizatis f metric spaces, Pacific J. Math. 11 (1961), B. Fitzpatrick, Jr., Sme tplgically cmplete spaces, Geeral Tbplgy ad its Applicatis 1 (1971), [G ] G. Gruehage, Stratifiable spaces are M2, Tplgy l Prceedigs 1 (1976), [G ], Stratifiable a-discrete spaces are M l, 2 Prc. AIDer. Math. Sc. 72 (1978), [He] [HJ] [HLZ] R. Heath, Stratifiable spaces are a-spaces, Ntices AIDer. Math. Sc. 16 (1969), 761. ad H. Juila, Stratifiable spaces as subspaces ad ctiuus images f M l -spaces (pre prit). R. Heath, D. Lutzer, ad P. Zer, Mtically rmal spaces, Tras. AIDer. Math. Sc. 178 (1973), [Hy] D. M. Hyma, A categry slightly larger tha the metric ad CW-categries, Mich. Math. J. 15 (1968),

29 104 Gruehage [J ] H. J. K. Jui1a, Neighbrets, Pacific J. Math (1968), [J ], Stratifiable spaces as subspaces ad c 2 tiuus images f M 1 -spaces, hadwritte tes. [M] T. Mizkami, O Nagata's prblem fr paracmpact a-metric spaces, Tplgy ad App1. II (1980), [N ] J. Nagata, A survey f the thery f geeralized 1 metric spaces, Prc. Third Prague Sympsium, 1971, [N ], O Hyma's M-space, Tplgy Cferece 2 VPI, Spriger-Verlag Lecture Ntes i Mathematics N. 375, [0] A. Okuyama, a-spaces ad clsed mappigs, I, Prc. [S] Japa Acad. 44 (1968), F. Slaughter, The clsed image f a metrizable space is M Prc. AIDer. Math. Sc. 37 (1973), , Aubur Uiversity Aubur, Alabama 36849