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 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.
3. Each open cover has a closure-preserving closed refinement.
4. Each open cover has an open refinement that is a countable union of closure-preserving families.

References

• Michael, Ernest (1953). "A note on paracompact spaces" (PDF). Proceedings of the American Mathematical Society. 4 (5): 831–838. doi:10.1090/S0002-9939-1953-0056905-8. ISSN 0002-9939. Archived (PDF) from the original on 2017-08-27.
• A. Mathew’s blog post

Further reading

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