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 of subsets of a topological space is said to be closure-preserving if for every subfamily ,

.

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

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

  1. 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.

References

  1. ^ Michael 1957, Theorem 1 and Theorem 2.
  2. ^ Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, ISBN 9780486131788, OCLC 829161886. Theorem 20.7.
  3. ^ Michael 1957, § 2.
  4. ^ Engelking 1989, Lemma 4.4.12. and Lemma 5.1.10.

Further reading

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