General topology Compactness Concept
Paracompact space
Summary
A paracompact space is a topological space in which every open cover has an open refinement that is a locally finite family. Every compact space and every metric space is paracompact. Paracompact Hausdorff spaces are normal, and they admit partitions of unity subordinate to any open cover. This is why paracompactness is a standard assumption on manifolds: it lets one glue local constructions into global ones.
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Further reading
About this page
| Tag | 8R29G6NG9Q |
|---|---|
| URI | m2://8R29G6NG9Q |
| Kind | concept |
| Writing | W0 stub |
| MSC | 54D20 |
| Specializes | Metacompact space |
| Same as | |
| Changed | 2 November 2026 |
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Used by
- Michael selection theorem
- Stone's theorem on paracompactness of metric spaces
- Dieudonné's theorem on paracompact Hausdorff spaces
- Dowker space
Results about this
- Stone's theorem on paracompactness of metric spaces
- Dieudonné's theorem on paracompact Hausdorff spaces