Collectionwise normal and Moore implies metrizable: Difference between revisions

Latest revision as of 02:48, 27 January 2012

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:

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

@@ Line 11: / Line 11: @@
 # If a [[topological space]] is both [[uses property satisfaction of::collectionwise normal space|collectionwise normal]] and a [[uses property satisfactin of::Moore space]], then it is a [[proves property satisfaction of::metrizable space]].
 # If a [[topological space]] is both [[collectionwise normal space|collectionwise normal]] and a [[uses property satisfaction of::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.