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Some Examples of Normal Moore Spaces Can. J. Math., Vol. XXIX, No. 1, 1977, pp. 84-92 SOME EXAMPLES OF NORMAL MOORE SPACES MARY ELLEN RUDIN AND MICHAEL STARBIRD 1. Introduction. A normal Moore space is non-metrizable only if it fails to be X-collectionwise normal for some uncountable cardinal X [1]. For each uncountable cardinal X we present a class Sf\ of normal, locally metrizable Moore spaces and a particular space Sx in $f\. If there is any space of class y?\ which is not X-collectionwise normal, then 5\ is such a space. The conditions for membership in j^x make a space in Sf\ behave like a subset of a product of a Moore space with a metric space. The class «y^x is sufficiently large to allow us to prove the following. Suppose F is a locally compact, 0-dimen- sional Moore space (not necessarily normal) with a basis of cardinality X and M is a metric space which is O-dimensional in the covering sense. If there is a normal, not X-collectionwise normal Moore space X where I C ^ X M, then Sx is a normal, not X-collectionwise normal Moore space. It is consistent with the usual axioms of set theory that there exist, for each uncountable cardinal X, a normal, not X-collectionwise Hausdorff Moore space [3]. It follows from the results quoted above that it is consistent that each Sx be a normal, non-metrizable Moore space. We also present, for each uncountable cardinal X, a locally metrizable Moore space T\ related to S\. If there is a first-countable, normal, not X-collectionwise normal space, then 7\ also fails to be X-collectionwise normal. P. Nyikos has shown that 7\ is normal if and only if it is metrizable, so 7\ itself cannot be an example of a normal, non-metrizable Moore space. Both S\ and 7\ are purely set theoretic in nature, being built from families of subsets of X. Either a proof that some Sx is not collectionwise normal or a proof that it is consistent that all 7\ are collectionwise normal (or, equivalently, normal) would settle the normal Moore space conjecture. 2. The class Sf\ and the normality lemma. The class 5f\ was defined as a result of investigation of normal Moore spaces which are subsets of a product of a Moore space with a metric space. Theorem 1 below gives a description of some of the spaces of this type which are in $f\. The conditions listed for membership in Sf\ are those which allow our proof of normality to work. Received March 3, 1976 and in revised form, October 1, 1976. The research of the first author was sponsored by NSF Grant MPS-73-08825, and that of the second author was sponsored by NSF Grant MPS-75-16158. 84 Downloaded from https://www.cambridge.org/core. 30 Sep 2021 at 04:02:17, subject to the Cambridge Core terms of use. NORMAL MOORE SPACES 85 Suppose that X is an uncountable cardinal. The class 5^x will contain all Moore spaces S for which the following hold. (1) There is a discrete family {Ca}a6x of closed sets in S. (2) D = S — UaexCe* is discrete. (3) If A C X, there are disjoint open sets U and V in 5 with \Ja(zACa C U and U/3ÉX-AC/3 C V. (4) There is a metric space M and, for each a £ X, a subspace ATa of M and homeomorphism ha from Afa onto Ca. (5) There is a clopen basis ^ for M such that ^ = U t&38 % where each 38 x is a discrete open cover of Af ; and, for each if w, there is a (discrete) family {0B|5 ^ 381\ oi open sets in 5 such that ha(B C\ Ma) C 0B and (_0B Pi Ca) Cha(B r\ Ma) for each a in X. (Note that B = B and Of? = Os. It is written in the preceding form for future reference.) If there is a normal, non-metrizable Moore space, there is one which has properties (l)-(3) above. If there is a locally metrizable, normal, non-me­ trizable Moore space, there is one which has properties (l)-(4) above. Property (5) is a contrived assumption which makes a space in Sf\ look enough like a subset of a product so that our proofs of normality and Theorem 3 below work. A somewhat larger class of spaces could be obtained by weakening (5) by assuming only that 38 is an open, rather than clopen, basis for M and that each 38 i is a discrete family of open sets which may not cover M. The proof below suffices to show that any space in this larger class is still normal. NORMALITY LEMMA. If S is in class Sf\, then S is normal. Proof. First note that it is sufficient to show that every pair of disjoint closed sets which lie in Uaexd can be separated by disjoint open sets. Let H and K be disjoint closed subsets of UaçxC*. We show that H and K can be separated by producing two countable collections { Ut} i€co and { V\\ ieœ of open sets such that H d U t^ Ui} K C U i& Vu and, for each i £ w, Z7* P\ K = 0 and Vtr\H = 0. Using 38 as described in (5), for each B £ 38 let HB = {a £ \\ha(B n Ma) r\H 9* 0 but ha(B H Ma) C\ K = 0}. Similarly let KB = {a £ \\ha(B P\ Ma) H K ^ 0 but ha(B H Ma) C\ H = 0}. By (3), there are disjoint open sets X# and YB in 5 with Ua<Ei/BC* C XB and Ua(EX-#£>Ca C YB. Also there are disjoint open sets ZB and 1/FB in 5 with Ua£KBCa C ZB and Ua€X-/S-BCa C Wfl. Let Ut = UB£®i(0B C\ XB) and Vt = \JB^m(0B C\ ZB) where 0B is de­ fined as in (5). Observe that { Ut) i€u covers H. For suppose p £ CaC\ H. There is an i G co and B £ 38t such that £ G ha(B H M«) but Aa(S H M«) H X = 0. Thus p £ 0B C\ XB C £7*. Similarly { F*} i€w covers K. Downloaded from https://www.cambridge.org/core. 30 Sep 2021 at 04:02:17, subject to the Cambridge Core terms of use. 86 M. E. RUDIN AND M. STARBIRD To show that ViCMI = 0, again assume p £ CaC\H. Since {0B\B ^ 38 ^ is discrete, there is at most one B <G 38 x with p £ ÔB. H p £ ha(B H Afa), then £ G WB and £ g ZB. Thus, since (ÔB H C«) CK(B C\ Ma), p Q Vt. Similarly K r\ ut = 0. Thus [/* and F* have the desired properties. 3. Some spaces in S^\. In the theorem below we describe a natural con­ struction which yields spaces in $/?\ and which, in fact, helped motivate the definition of 5^\. THEOREM 1. Let Y be a Moore space with a discrete family {Ca'}a£\ of closed r sets such that each Ca is compact and 0-dimensional. Let M' be a metric space which is ^-dimensional in the covering sense. Let X' be a sub space of Y X M'. Let X be the space obtained from X' by making ((F — UaçxC*) X M') H X' discrete. Then if X is normal, X belongs to 3/\. Proof. Since X is a Moore space, we proceed to check that X satisfies proper­ ties (l)-(o). Each Ca in (1) is (CJ X M')r\X. The D in (2) equals ((Y — Ua&Ca) X M') H X. Property (3) is guaranteed by the normality of X. Let E be the Cantor set. Then the M of (4) is £ X M'. Note that each Cd is a compact, O-dimensional metric space; hence there is a subset Ea of E and a homeomorphism ha' from Ea onto CJ. Then ha" = ha' X idM> : Ea X M' —» l f C«' X M' is a homeomorphism. Now Ma in (4) is (ha")- (X C\ (Ca X Ma')) and ha in (4) is ha" restricted to Ma. Let 38' — \J i^SS { be a nested basis for E so that each 38 ( is a discrete open cover of E. Let 38" = \J i&>38 " be such a basis for M'. Each «a?* in (5) equals \W X B"\B' £ 38( and 5" G 381'\. And ^ = Ui^i will be the basis of M = E X M' required in (5). ni For each i G w, let 38 ( = {Bi/} j=i . By the normality of X, there are dis­ ni joint open sets {Oi/}j=i in X so that for each j, UaA((^i/ X M') C\ Ma) C Otj'. Let 0BlJ>XB>, = Oi/ r\ (Y X B"). This collection of O's satisfies (5). 4. The description of S\. In this section we describe the space 5\ in Sf\ which is a normal, non-metrizable Moore space if any space in S^\ is such a space. (See Theorem 3 below.) Assume that X is an uncountable cardinal. We think of 22 as the set of all collections of subsets of X. We think of (22 )x as the set of all functions from X into 22 . The metric space M associated with the space 5\ is obtained by taking the product (with the product topology) of co copies of (22 )x with the discrete topology.
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