Michael's theorem on paracompact spaces

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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:^[1]

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

$X$ is paracompact.
Each open cover has a closure-preserving refinement, not necessarily open.
Each open cover has a closure-preserving closed refinement.
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.

Notes

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

References

Michael, E. (1957), "Another note on paracompact spaces", Proceedings of the American Mathematical Society, 8 (4): 822–828, doi:10.1090/S0002-9939-1957-0087079-9, JSTOR 2033306, MR 0087079
Mathew, Akhil (August 19, 2010), "A theorem of Michael on paracompactness", Climbing Mount Bourbaki
Engelking, Ryszard (1989), General Topology, Sigma Series in Pure Mathematics, vol. 6 (2nd ed.), Berlin: Heldermann Verlag, ISBN 3-88538-006-4, MR 1039321

Statement

Proof sketch

Notes

References

Further reading