Michael's theorem on paracompact spaces

Template:Short description 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:

Frequently, the theorem is stated in the following form:

In particular, a regular-Hausdorff Lindelöf space is paracompact. The proof of the theorem uses the following result which does not need regularity:

Proof sketch

Template:Expand section The proof of the proposition uses the following general lemma

References

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Further reading

• Michael's Theorem in Ncatlab