Normal Moore Space Research Articles

Screenability, Pointwise Paracompactness, and Metrization of Moore Spaces

F. B. Jones (6) has shown that, if , then every separable normal Moore space is metrizable. It is not known whether this assumption is necessary, though perhaps some progress is made in (5). However, it is easily seen from R. H. Bing's Example E in (3) that a certain condition (see (1) below) implied by is necessary. Also in (3), Bing showed that every screenable normal Moore space is metrizable. In this paper we establish that: (1) every separable normal Moore space is metrizable if and only if every uncountable subspace M of E1 contains a subset which is not an Fσ (in M); (2) if every pointwise paracompact normal Moore space is metrizable, then so is every separable normal Moore space; (3) every screenable Moore space is pointwise paracompact but not conversely; (4) a T3-space is a pointwise paracompact Moore space if and only if it has a uniform base (in the sense of (1, p. 40), not a uniformity).

A note on complete collectionwise normality and paracompactness

1. A question that has aroused considerable interest and which has remained unanswered is the following. Is a normal Moore space metrizable? Both R. H. Bing and F. B. Jones have important results which go a long way toward answering this question. For example, Bing has proved that a collectionwise normal Moore space is metrizable [l ] while Jones has shown that a separable normal Moore space is metrizable [2] provided the continuum hypothesis holds true. Although the results given in the present paper fail to answer the question at hand, it is shown that certain types of normality and paracompactness are equivalent in a Moore space, indeed, in more abstract spaces such as a semimetric topological space.2 2. Complete collectionwise normality. The reader is referred to R. H. Bing's paper [l] for definitions concerning screenability, collectionwise normality, discrete collections, and related concepts. We say that a collection G is almost discrete if and only if it is discrete with respect to G* (the logical sum of the elements of G). Furthermore, a space is completely collectionwise normal if and only if for each almost discrete collection G of point sets, there is a collection H of disjoint open sets covering G* such that no element of H intersects two elements of G. A space is hereditarily collectionwise normal if and only if each subspace is collectionwise normal. It is not difficult to show that complete collectionwise normality is equivalent to hereditary collectionwise normality in a topological space. A topological space 5 is said to be F,-screenable if and only if for each open covering G of S, there exists a sequence {Xi} such that (1) for each i, Xi is a discrete collection of closed point sets each of which lies in an element of G and (2) E-^«' covers S. There is an example [5] of a Hausdorff space satisfying the first axiom of count