Metrizability of certain countable unions
H. H. Corson, E. Michael · Illinois Journal of Mathematics · 1964
IntroductionSuppose the regular space X is the union of a collection of metrizable subsets.A number of conditions for X to be metrizable under these cir- cumstances are known, but in all of them the elements of 9 are either open [8], [10], or closed [7], [10], or separable with X compact [9].In this paper we consider the case where 9 is countable, and where each M 9r is a dense subset of-an open set; such sets M will be called locally dense.Now let the regular space X be the union of a countable collection i) of metrizable, locally dense subsets Mn.As Example 6.5 shows, these assump- tions alone do not imply that X is metrizable, even if has only two ele- ments.Further conditions are needed to insure metrizability, and they fall into two classes: On the one hand, our assumptions imply that X has a point-countable base, and this has two immediate consequences.First, if X is separable, it has a countable base and is therefore metrizable; second, if X is compact, it is metrizable by a theorem of A. Mishchenko [6].On the other hand, it will be shown that X is metrizable if it is normal and there are generalized F (in X) sets An c Mn which cover X.Some of the principal consequences of these facts are summarized in the following theorem.THEOREM 1.1.If the normal space X is the union of a countable collection 9 of locally dense, metrizable subsets Mn, then X is metrizable if it satisfies any of the following conditions:(a) X is separable (in particular, each Mn is separable).(b) X is locally compact.c X is a-compact.d Every open set in X is an F. e There exist F,-sets A,, c M,, which cover X.