Metacompact space

Abstract: In 1958 M. Katetov proved that in a normal space X, X is expandable if and only if X is collectionwise normal and countably paracompact. This result has since been used to answer many questions in various areas of general topology. In this paper Katetov's theorem is generalized for nonnormal spaces and various characterizations of collectionwise normality are shown. Results concerning metrization, paracompactness, sum theorems, product theorems, mapping theorems and M-spaces are then obtained as applications of these new theorems. Introduction. In [14] L. Krajewski investigated the property of expanding locally finite collections to open locally finite collections and obtained various results relating this property with certain topological covering properties, metrization theorems, sum theorems and product theorems. In ?1 we summarize the known results concerning expandability, collectionwise normality and these topological covering properties; and then we show that every metacompact space is almost expandable. In ?2 we introduce various generalizations of the notion of expandability and give characterizations for collectionwise normality. The theorem of Katetov [13] (Theorem 1.4 below) is then generalized. Characterizations of expandability properties in terms of open covers are given in ?3; and in ?4 we characterize the properties of paracompactness, subparacompactness, and metacompactness and establish equivalences in certain expandable spaces. Various mapping theorems are proved in ?5, and product theorems are obtained in ?6. We obtain results cQncerning subspaces and sum theorems in ?7. As applications of the previous results, metrization theorems are observed in ?8. Examples and unanswered questions are given in ?9. REMARK. A space X is subparacompact (Fa-screenable) if every open cover has a a-discrete closed refinement. The symbol IAI will be used to denote the cardinality of the point set A. A space X will be a T1 topological space unless otherwise stated. 1. Preliminary results. The following definition is due to L. Krajewski [14]. DEFINITION 1.1. A space X is called m-expandable, where m is an infinite cardinal, if for every locally finite collection {F< a E A} of subsets of X with IAI < m, then Presented to the Society, January 24, 1971; received by the editors June 11, 1970 and, in revised form, November 3, 1970. AMS 1969 subject classifications. Primary 5423; Secondary 5435, 5450.

Abstract: It is shown that the normal T1-space X is realcompact if and only if (a) each discrete subset of X is realcompact and (b) X can be embedded as a closed subset in the product of a collection of regular 0-refinable spaces. We will say that a space X has property (*) if it is true that each discrete subset of X is realcompact; i.e., the cardinality of each discrete subset of X is nonmeasurable. In [5], the author has shown that a normal T1-space X is realcompact if and only if X has property (*) and X can be embedded as a closed subspace in the product of a collection of subparacompact spaces and metacompact spaces. S. Mrowka suggested to the author that there should be a nontrivial class of spaces 9 containing the class of subparacompact spaces and the class of metacompact spaces so that a normal space X is realcompact if and only if X has property (*) and X can be embedded as a closed subspace in a product of members of bY. It is the purpose of this paper to show that the class of 0-refinable spaces, introduced by Worrell and Wicke in [4], is such a class. Recall that a space X is 0-refinable if it is true that if r is an open cover of X then there is a sequence Y'1, '2, * * * of open covers of X that refine Y" such that if x E X, then there is an integer i such that only finitely many members of 1l, contain x. Clearly, any metacompact space is 0-refinable. It is shown in [1] that any subparacompact space is 0-refinable. Our notation will follow that of [2]. LEMMA 1 [5]. Suppose that X is a Tl-space and 6' is a class of T73spaces such that the topology on X is the weak topology induced by C(X, A). Then X can be embedded as a closed subspace in the product of a collection of members of 6' if and only if it is true that if Y is a free ultrafilter of closed subsets of X, then there are a memberf of C(X, t) and an open cover I4 of range (f) such that { f-1(U): U E &} refines {(X-F): F E S}. LEMMA 2 (THEOREM 18, [3]). If Oil is an open cover of the space X, then Received by the editors January 15, 1973. AMS (MOS) subject class;fications (1970). Primary 54D60, 54D20.