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

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

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

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

1. $X$ 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 —  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 —  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 — 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.

References

• Ernest Michael, Another note on paracompactness, 1957
• A. Mathew’s blog post
• Ryszard Engelking, General Topology, Revised and Completed Edition, Heldermann Verlag, Berlin, 1989.

Further reading

• Michael's Theorem in Ncatlab