Michael's theorem on paracompact spaces
In mathematics, Michael's theorem gives sufficient conditions for a regular topological space to be paracompact.
Statement
A family of subsets of a topological space is said to be closure-preserving if for every subfamily,For example, a locally finite family of subsets has this property. With this terminology, the theorem states:
Frequently, the theorem is stated in the following form:
In particular, a regular-Hausdorff Lindelöf space is paracompact. The proof of the theorem uses the following result which does not need regularity: