In mathematics, Michael's theorem gives sufficient conditions for a regular topological space (in fact, for a T1-space) to be paracompact.
Statement.
A family formula_1 of subsets of a topological space is said to be closure-preserving if for every subfamily formula_2,
formula_3.
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:
Proof sketch.
The proof of the proposition uses the following general lemma