A NOTE ON NORMALITY AND COLLECTIONWISE NORMALITY

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1 Volume 7, 1982 Pges A NOTE ON NORMALITY AND COLLECTIONWISE NORMALITY by Frnklin D. Tll Topology Proceedings Web: Mil: Topology Proceedings Deprtment of Mthemtics & Sttistics Auburn University, Albm 36849, USA E-mil: topolog@uburn.edu ISSN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume A NOTE ON NORMALITY AND COLLECTIONWISE NORMALITY Frnklin D. Tll l In this note we mke severl unrelted observtions concerning circumstnces under which normlity implies collectionwise normlity. All spces re ssumed Husdorff. I. Screenble Spces In [Tll I proved tht screenble norml spces re collectionwise Husdorff. I cn now improve this. Theorem 1. Screenble norml spces re collection ~ise norml ~ith respect to countbly metcompct closed sets. Proof. Let mbe discrete collection of countbly metcompct closed sets. Let 5 = u < 5 n w n be -disjoint refinement of the cnonicl cover. The closed subspce Um is norml nd countbly metcompct. Since point-finite open covers of norml spces cn be shrunk, there re open {Tn}n<w' Tn C u5 n um. Um is closed so the closure sign n is unmbiguous. {M n T : M E m} for ech n is discrete n collection seprted by the open sets Sn,M = U{S E 5n: s n M ~ ~}. By normlity the S MiS my be shrunk to n, discrete seprtion {S' M: M Em}. Then U{S' : M E m} is n,l: n,m -discrete nd so yields seprtion by stndrd rguments. Theorem 1 cnnot be improved in ZFC to get collectionwise normlity; ssuming 0++, M. E. Rudin [R l ] constructed 1 The uthor cknowledges support from Grnt A-7354 of the Nturl Sciences nd Engineering Reserch Council of Cnd.

3 268 Tll screenble norml spce which is not countbly metcompct. Assuming there is such spce, in [R ] she then produces 2 screenble norml non-collectionwise norml spce. II. Getting By With Less In this section we give severl exmples of how results tht re obvious if Nyikos' "Product Mesure Extension Axiom" [N I ] is ssumed, cn be obtined from weker set-theoretic ssumptions with bit more topology. The crucil ingredient is the following theorem, which is due to the uthor nd w. Weiss [TW]. Also see W. Fleissner [F ]. Somewht weker 6 results were erlier obtined by Crlson [C]. In the first version of this note I used his work nd the mesure-extension techniques of Nyikos to obtin collectionwise normlity results. It hs since become cler tht these methods re unnecessry nd tht stronger results my be obtined directly. Some of the mesure extension proofs still pper in [T ]. 4 Theorem 2. Adjoin p Cohen (rndom) rels to model of set theory. If X is norml nd Y is discrete collection such tht luyi < cofinlity of p nd ech point of uy hs chrcter less thn p, then X is collectionwise norml with respect to Y. If p is wekly compct, it suffices to hve I uyi ~ p. Using this theorem, we obtin the following result: Theorem 3. Adjoin p mny Cohen (rndom) rels to model of V = L, P regulr. Then norml spces of chrcter less thn pre collectionwise norml with respect to

4 TOPOLOGY PROCEEDINGS Volume discrete cozzections of sets of crdinzity Zess thn p. Proof. By theorem 2 we cn tke cre of collections with unions of size less thn p, so in prticulr, X is <p-collectionwise Husdorff. If we collpse set of crdinlity less thn p in spce of chrcter less thn p to point, tht resulting point in the quotient spce hs chrcter 2<P, which in this model is just p. In L[A), A : p, GCH holds t p nd bove, while 0 for sttionry systems holds for regulr crdinls ~p. By [F ), norml l spces of chrcter ~p which re <p-collectionwise Husdorff re then collectionwise Husdorff. The Theorem follows. In [F ) Fleissner proved tht.in the model obtined by 2 Levy-collpsing n inccessible crdinl over model of V = L, the chrcter of copies of WI in first countble spces ws ~l' nd (hence) tht in norml first countble spces, discrete collections of copies of WI could be seprted. In [T ) I observed tht the ltter result could 2 be obtined without n inccessible by djoining ~3 Cohen subsets of WI with countble conditions, collpsing ~3 to ~2 by conditions of size ~~l' nd then dding K+ Cohen subsets of K for ll regulr K ~ K 2 by reverse Eston forcing. It follows from the previous Theorem tht djoining ~2 Cohen (rndom) rels to model of V = L will lso do the trick. The following topologicl lemm will enble us to extrct more results from Theorems 2 nd 3. This lemm is essentilly proved in [B), but we give the proof for the reder's convenience.

5 270 Tll Lemm 4. If X is hereditrizy ozzetionwise 'Husdorff nd eh point of X hs neighbourhood of weight <K~ X is the topozogiz sum of spes of weight ~K. Proof. open sets of weight <K. defined to be unions of ~K Let U o be mximl disjoint collection of then Suppose US,S <, < K hve been mny disjoint collections of open sets of weight <K. Let F X - US<UUS. Let V be collection of open subsets of X of weight <K such tht {F n V: V V } is mximl disjoint collection of rel tively open subsets of F- Fix dense set D V of power <K in ech F n V. Any selection of points, one from ech D v ' yields set which is closed nd discrete in uv, which my then be seprted by disjoint open sets of weight <K. my therefore cover U{D V : V V} by collection U of open sets of weight <K, such tht V is the union of <K collections of disjoint open sets. Clim X = U < uv Suppose K x ~ U<KuV. Let W be neighbourhood of x of weight <K. since the F'S re descending nd W hs hereditry Lindelof number Then there is n < K such tht F n W = F + l n w, <K_ Then UU + l n W is empty so F + l n W is empty, becuse uu + 1 n F + l is dense in F + l - since x n<kf- We But tht's contrdiction U < U is the union of _<K mny collections of disjoint K open sets. Ech member of U < U intersects fewer thn K K mny elements of ech such collection. It follows by stndrd rguments tht X is the sum of subspces, ech of which is the union of <K open sets of weight <K_ subspce then hs weight <K. Ech such

6 TOPOLOGY PROCEEDINGS Volume The fruit of these results is Theorem S. Adjoin K++ Cohen (rndom) rets to modet of V L. Then hereditrity normt spces of toct weight <K re hereditrity cottectionwise normt. Proof. Since chrcter is ~ locl weight, by Theorem 3 such spce X is hereditrily collectionwise Husdorff. Theorem 4 it decomposes into subspces of weight ~K+ nd By then X is collectionwise norml by Theorem 3 or 2. Theorem 6. Adjoin wekty compct mny Cohen (rndom) rets to modet of V = L. Then hereditrity normt spces of toct weight <2 NO re hereditrity cottection~ise normt. Of course by mking the significntly stronger ssumption of the consistency of the existence of strongly compct crdinl, there is model where "hereditrily" my be omitted in both plces nd locl weight replced by chrcter. Tht is the originl Kunen-Nyikos result for rndom rels [K] (or see [F S ])' [NIl. L does not enter the picture in tht cse. For the non-logicins, we note tht the consistency of the existence of wekly compct crdinl implies tht such crdinl consistently exists in L (see e.g. [D]), nd hence tht tht mny rels my be djoined to L. To prove the Theorem, s before the spce decomposes into the sum of clopen pieces, ech of which is the union NO NO of <2 sets, ech of which hs weight <2 nd hence No N No crdinlity <2, since in this model 2<2 O 2 Hence NO ech piece hs crdinlity ~2, so by Thebrem 2 we re done.

7 272 Tll Theorem 7. Adjoin wekzy compct mny Cohen (rndom rezs) to modez of set theory. Then normz ZocZZy con NO nected spces of chrcter <2 re cozzectionwise normz if they toctty hve crdintity <2~O. Note tht locl cellulrity or locl Lindelof number NO NO <2 NO NO <2 yields locl crdinlity <2 since 2 2 here. (For the cse of rnnifolds, this result is due to Nyikos [N 2 ]. To prove the Theorem it suffices to look t ech component. But by n rgument of Reed nd Zenor NO [RZ], ech component hs crdinlity 2 I conjecture tht norml mnifolds re consistently collectionwise norml without lrge crdinls. I cn get this result for norml mnifolds which e.g. hve cellulrity ~ N I or in which ech closed set is the intersection of ~ N I open sets, by closer nlysis of the proof of Lemm 4. In [T ] I proved tht N rndom rels djoined to L 3 2 ensure tht every loclly compct perfectly norml spce is collectionwise norml. Restricting in different direction, we hve of V = L. Theorem 8. Adjoin N Cohen (rndom) rezs to modez 2 Then normz mnifozds which re oe-refinbze re cozzectionwise normz. Recll oe-refinbility is simultneous generliztion of e-refinbility nd metlindelofness, nmely every open cover hs n open refinement which is the union of countbly mny covers, such tht for ech point there is n n such

8 TOPOLOGY PROCEEDINGS tht it's in only countbly mny members of the n'th cover. 8-refinble norml mnifolds re collectionwise norml by stndrd rguments: since they're loclly developble, they're developble [WW] nd hence perfectly norml; since they're loclly compct, loclly connected, nd perfectly norml, they're collectionwise norml with respect to compct sets [AZ]i since they're 8-refinble, loclly compct, nd col~ lectionwise norml with respect to compct sets, they're prcompct. In fct, by more difficult rgument, 8-refinble norml, loclly compct, loclly connected spces re prcompct [G]. Metlindelof mnifolds re collectionwise norml by n even esier rgument: they're loclly seprble nd metlindelof, so prcompct. We prove the Theorem by blend of [RZ] nd [AP]. It suffices to show tht o8-refinble norml loclly second countble connected spces hve weight ~ N l We cn do this if we cn construct second countble open sets {U } < such wi For then U < U = U < U, which by first wi wi countbility is closed, so U < U = X. But then the union wi of the bses for the U's is bsis for the whole spce. It suffices to prove tht the closure of second countble open set is Lindelof for then, given U, we cover U by second countble open sets {Vn}n<w' nd let U + l = Un<wVn. Suppose then tht U is open nd second countble, but U is not Lindelof. U is o8-refinble, so it hs n uncountble closed discrete subspce [Au]. But U is seprble, norml, nd by Theorem 2, Nl-collectionwise Husdorff, so it cnnot hve such subspce.

9 274 Tll III. ObservtioDs OD Theofem ofshelh Fleissner [F 3 ], [F 4 ] nd Shel~ [S] hve investig~ed the question of under wht circumstnces does ~l-collectionwise Husdorff imply collectionwise Husdorff. For exmple, Shelh proves tht in the model obtined by Levy-collpsing supercompct crdinl to w 2 ' n Nl-collectionwise Husdorff spce is collectionwise Husdorff if it is loclly countble (i.e. ech point hs neighbourhood of crdinlity ~ ~O). Similr questions 'cn be sked for collectionwise normlity. nopml~ Theopem 9. We cn prove V = L implies tht if X is hepeditpily (hepeditpilyj collectionwise nopml with pespect to discpete collections of ~ N l sets which pe ech of pdinlity ~ ~l~ loclly hepeditpily Lindel3f~ nd loclly hspeditpily seppble~ then it is (hepeditpilyj coz'lectionbjise nopml. Ppoof. Since s~prble regulr spces hve weight NQ <2 (see e.g. [J), X hs chrcter < N I, so it is hereditrily collectionwise Husdorff. By the proof of Lemm 4, without loss of generlity X my be ssumed to be the union of ~~l hereditrily Lindelof subspces. Heredi NQ trily Lindelof regulr spces hve crdinlity <2 (see e.g. [Ju]), so by CH nd hypothesis, X is collectionwise nq~l. All properties re hereditry, so we lso hve the hereditry version. More in the spirit of Shelh's results we hve

10 TOPOLOGY PROCEEDINGS Volume Theorem 10~ In the model obtined by L~vy-ollp8ing superompt to w2~ if X is lolly countble spe whih hereditrily is olletionwise norml with respet to disrete olletions of ~ ~l set8~ eh of rdinlity < ~l~ then X is hereditrily olletionwise norml. Proof. By hypothesis X is hereditrily ~l-collectionwise Husdorff. By Shelh X is hereditrily collectionwise Husdorff. By the proof of Lemm 4, X decomposes into the sum of subspces of crdinlity ~~l' so by hypothesis it is hereditrily collectionwise norml. Shelh's method works for spces stisfying somewht less stringent conditions thn locl countbilityi thus Theorem 10 cn be improved. However, since the detils of his rgument do not pper in [S], it would tke us too fr field to develop them here. References [AP] K. Alster nd R. Pol, Moore spe8 nd olletionwise Husdorff property, Bull. Acd. Polon. Sci. Sere [Au] [B] [C] Sci. Mth. Astronom. 23 (1975), C. E. Aull, A generliztion of Theorem of Aquro, Bull. Austrl. Mth. Soc. 9 (1973), Z. Blogh, On scttered spces nd loclly nice spces under Mrtin's Axiom, Co~~ent. r1th. Univ. Croline (to pper). T. Crlson, Extending Lebesgue mesure by countbly mny sets, Pc. J. Mth. (to pper) [0] K. J. Devlin, A8pets of onstrutibility, Lect. Notes Mth. 354, Springer-Verlg, Berlin, [F l ] W. G. Fleissner, Norml Moore spes in the onstrutible universe, Proc. Amer. Mth. Soc. 46 (1974),

11 276 Tll [F 2 ], The ch~cte~ of wi in fi~st countble spces, Proc. Amer. Mth. Soc. 62 (1977), [F 3 ], On A collection Husdo~ff spces, Top. Proc. 2 (1977), [F 4 ], An ~iom fo~ nonsep~ble Bo~el theo~y, Trns. Amer. Mth. Soc. 251 (1979), [F S ], The norml Moore spce,conjectu~e nd l~ge crdinls, Hndbook of Set-theoretic Topology, ed. K. Kunen nd J. Vughn, North-Hollnd, Amsterdm (to pper) [F 6 ], Lyn~es, strongly compct crdinls, nd the [G] [J] Kunen-Pris technique (preprint). G. Gruenhge, Prcompctness in nopml, loclly connect,ed, locll,y compct spces, Top. Proc. 4 (1979), I. Juhsz, Crdinl, functions--ten yers ftsp, Mth. Centre, Amsterdm, [K] K.~ Kunen, Mesures on 2 A, hndwritten mnuscript. [N ] P. J. Nyikos, A ppovisionl, sol,ution to the norml, 1 Moore spce probl,em, Proc. Amer. Mth. Soc. 78 (1980), [N ] 2, Set-theopetic topol,ogy of mnifolds [R 1 ] M. E. Rudin, A screenbl,e norml, non-prcompct spce, Top. Appl. 15, 3" [R 2 ], CoZZection~i8e no~mzity in sc~eenbze 8pces (preprint). [RZ] G. M. Reed nd P. L. Zenor, Metpiztion of geneplized mnifolds, Fund. Mth. 91 (1976), [S] S. She1h, Remrks on A-coll,ection~ise Husdopff spces, TOp. Proc. 2 (1977), [T ] F. D. Tll, Set-~~eo~etic consistency pesults nd 1 topol,ogicl, theorems concerning the norml, Moo~e spce conjectupe nd ~elted probl,ems, Thesis, University of Wisconsin, Mdison, 1969; Dissert. Mth 148 (1977), 1-53.

12 TOPOLOGY PROCEEDINGS Volume [T ], Some pplictions of smll crdinl collpse 2 in topology, in Proc. Bolyi Jnos Mth. Soc Colloquium on Topology, Budpest, [T ], Collectionwise normlity without lrge 3 crdinls, Proc. Amer. Mth. Soc. 85 (1982), [T ], Normlity versus collectionwise normlity, 4 Hndbook of Set-theoretic Topology, ed. K. Kunen nd [TW] [WW] J. Vughn, North-Hollnd, Amsterdm (to pper). nd W. A. R. Weiss, A new proof of the consistency of the norml Moore spce conjecture (in preprtion). J. Worrell nd H. Wicke, Chrcteriztions of developble topologicl spces, Cn. J. Mth. 17 (1965), University of Toronto Toronto, Ontrio, Cnd