Michael's theorem on paracompact spaces

In mathematics, Michael's theorem gives necessary and sufficient conditions for a regular topological 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.

The corollary immediately implies that a regular-Hausdorff Lindelöf space is paracompact.

References

• Ernest Michael, Another note on paracompactness, 1957
• A. Mathew’s blog post

Further reading

• https://ncatlab.org/nlab/show/Michael%27s+theorem