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 $E_i$ 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:{{harvnb|Michael|1957|loc=Theorem 1 and Theorem 2.}}

{{math_theorem|math_statement=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:

{{math_theorem|name=Corollary|math_statement={{citation|title=General Topology|series=Dover Books on Mathematics|first=Stephen|last=Willard|publisher=Courier Dover Publications|year=2012|isbn=9780486131788|oclc=829161886}}. Theorem 20.7. 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:

{{math_theorem|name=Proposition|math_statement={{harvnb|Michael|1957|loc=§ 2.}} 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

{{math_theorem|name=Lemma|math_statement={{harvnb|Engelking|1989|loc=Lemma 4.4.12. and Lemma 5.1.10.}} 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

References

{{citation

| last = Michael | first = E.

| doi = 10.1090/S0002-9939-1957-0087079-9

| journal = Proceedings of the American Mathematical Society

| jstor = 2033306

| mr = 87079

| pages = 822–828

| title = Another note on paracompact spaces

| volume = 8

| year = 1957| issue = 4

}}

{{citation|first=Akhil|last=Mathew|url=https://amathew.wordpress.com/2010/08/19/a-theorem-of-michael-on-paracompactness/|title=A theorem of Michael on paracompactness |work=Climbing Mount Bourbaki|date=August 19, 2010}}
{{citation

| last = Engelking | first = Ryszard

| edition = 2nd

| isbn = 3-88538-006-4

| mr = 1039321

| publisher = Heldermann Verlag | location = Berlin

| series = Sigma Series in Pure Mathematics

| title = General Topology

| volume = 6

| year = 1989}}

Michael's theorem on paracompact spaces

Statement

Proof sketch

Notes

References

Further reading