Collectionwise normal and Moore implies metrizable

This article is about a metrization theorem: a theorem that gives necessary and sufficient conditions for a metric (possibly with additional restrictions) to exist. In particular, it gives some conditions under which a topological space is metrizable.

History

This statement was proved by Moore.

Statement

The statement has the following equivalent forms:

1. If a topological space is both collectionwise normal and a Moore space, then it is a metrizable space.
2. If a topological space is both collectionwise normal and a developable space, then it is a metrizable space.

Note that the converse is easily seen to be true: metrizable implies Moore and metrizable implies collectionwise normal. Thus, the above give necessary and sufficient conditions for a topological space to be metrizable.