Paracompact Hausdorff space
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
This article describes a property of topological spaces obtained as a conjunction of the following two properties: paracompactness and Hausdorffness
Definition
A topological space is termed paracompact Hausdorff if it satisfies the following equivalent conditions:
- It is paracompact and Hausdorff
- Given any open cover of the space, there is a partition of unity subordinate to that open cover; in other words, there is a partition of unity such that the support of each function is contained in some set of that open cover
- It is regular and every open cover has a locally finite open refinement
- It is regular and every open cover has a locally finite closed refinement
- It is regular and every open cover has a locally finite refinement
- It is regular and every open cover has a countably locally finite open refinement
The second definition is the one used in algebraic topology.
Relation with other properties
Stronger properties
- Compact Hausdorff space
- Locally compact paracompact Hausdorff space
- Polyhedron
- CW-space: For full proof, refer: CW implies paracompact Hausdorff
- Metrizable space
- Manifold