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| {{Short description|Theorem in topology}} | |||
| In mathematics, '''Michael's theorem''' gives |
In mathematics, '''Michael's theorem''' gives sufficient conditions for a ] ] (in fact, for a ]) to be ]. | ||
| == Statement == | == Statement == | ||
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| Frequently, the theorem is stated in the following form: | Frequently, the theorem is stated in the following form: | ||
| {{math_theorem|name=Corollary|math_statement=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.}} | {{math_theorem|name=Corollary|math_statement=<ref>{{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.</ref> 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 ] is paracompact. The proof of the theorem uses the following result which does not need regularity: | |||
| {{math_theorem|name=Proposition|math_statement=<ref>{{harvnb|Michael|1957|loc=§ 2.}}</ref> Let ''X'' be a ]. If ''X'' satisfies property 3 in the theorem, then ''X'' is paracompact.}} | |||
| == Proof sketch == | |||
| {{expand section|date=December 2024}} | |||
| The proof of the proposition uses the following general lemma | |||
| {{math_theorem|name=Lemma|math_statement=<ref>{{harvnb|Engelking|1989|loc=Lemma 4.4.12. and Lemma 5.1.10.}}</ref> 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 == | == References == | ||
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| * Ernest Michael, , 1957 | * Ernest Michael, , 1957 | ||
| * A. Mathew’s | * A. Mathew’s | ||
| * Ryszard Engelking, General Topology, Revised and Completed Edition, Heldermann Verlag, Berlin, 1989. | |||
| == Further reading == | == Further reading == | ||
| * https://ncatlab.org/nlab/show/Michael%27s+theorem | * | ||
| {{topology-stub}} | {{topology-stub}} | ||
| ] | |||
Latest revision as of 16:27, 4 January 2025
Theorem in topologyIn 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 ,
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:
Theorem — Let be a regular-Hausdorff topological space. Then the following are equivalent.
be a regular-Hausdorff topological space. Then the following are equivalent.
- 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 — 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 — Let X be a T1-space. If X satisfies property 3 in the theorem, then X is paracompact.
Proof sketch
 | This section needs expansion. You can help by adding to it. (December 2024) |
The proof of the proposition uses the following general lemma
Lemma — 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
- Michael 1957, Theorem 1 and Theorem 2. harvnb error: no target: CITEREFMichael1957 (help)
- Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, ISBN 9780486131788, OCLC 829161886. Theorem 20.7.
- Michael 1957, § 2. harvnb error: no target: CITEREFMichael1957 (help)
- Engelking 1989, Lemma 4.4.12. and Lemma 5.1.10. harvnb error: no target: CITEREFEngelking1989 (help)
- Ernest Michael, Another note on paracompactness, 1957
- A. Mathew’s blog post
- Ryszard Engelking, General Topology, Revised and Completed Edition, Heldermann Verlag, Berlin, 1989.
Further reading
 | This topology-related article is a stub. You can help Misplaced Pages by expanding it. |