Michael's theorem on paracompact spaces

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 ${\displaystyle E_{i}}$ of subsets of a topological space is said to be closure-preserving if for every subfamily ${\displaystyle E_{i_{j}}}$,

${\displaystyle {\overline {\bigcup E_{i_{j}}}}=\bigcup {\overline {E_{i_{j}}}}}$.

For example, a locally finite family of subsets has this property. With this terminology, the theorem states:^[1]

Theorem—Let ${\displaystyle X}$ be a regular-Hausdorff topological space. Then the following are equivalent.

1. ${\displaystyle X}$ is paracompact.
2. Each open cover has a closure-preserving refinement, not necessarily open.
3. Each open cover has a closure-preserving closed refinement.
4. Each open cover has a refinement that is a countable union of closure-preserving families of open sets.

Frequently, the theorem is stated in the following form:

Corollary—^[2] A regular-Hausdorff topological space is paracompact if and only if each open cover has a refinement that is a countable union of locally finite families of open sets.

In particular, a regular-Hausdorff Lindelöf space is paracompact. The proof of the theorem uses the following result which does not need regularity:

Proposition—^[3] Let X be a T₁-space. If X satisfies property 3 in the theorem, then X is paracompact.

Proof sketch

The proof of the proposition uses the following general lemma

Lemma—^[4] Let X be a topological space. If each open cover of X admits a locally finite closed refinement, then it is paracompact. Also, each open cover that is a countable union of locally finite sets has a locally finite refinement, not necessarily open.