Paracompact Hausdorff space

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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:

  1. It is paracompact and Hausdorff
  2. 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
  3. It is regular and every open cover has a locally finite open refinement
  4. It is regular and every open cover has a locally finite closed refinement
  5. It is regular and every open cover has a locally finite refinement
  6. 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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
compact Hausdorff space compact and Hausdorff compact implies paracompact paracompact Hausdorff not implies compact |FULL LIST, MORE INFO
locally compact paracompact Hausdorff space paracompact Hausdorff and locally compact |FULL LIST, MORE INFO
polyhedron the underlying topological space of a simplicial complex CW-space|FULL LIST, MORE INFO
CW-space the underlying topological space of a CW-complex CW implies paracompact Hausdorff |FULL LIST, MORE INFO
metrizable space the underlying topological space of a metric space metrizable implies paracompact Hausdorff Elastic space|FULL LIST, MORE INFO
manifold

Weaker properties

Property Meaning Proof of implication proof of strictness (reverse implication failure) intermediate notions
binormal space product with the unit interval is normal paracompact Hausdorff implies binormal |FULL LIST, MORE INFO
normal space any two disjoint closed subsets are separated by disjoint open subsets paracompact Hausdorff implies normal normal not implies paracompact Binormal space|FULL LIST, MORE INFO
normal Hausdorff space normal and a Hausdorff space paracompact Hausdorff implies normal normal not implies paracompact |FULL LIST, MORE INFO
Collectionwise normal space any discrete collection of closed subsets can be separated by pairwise disjoint open subsets paracompact Hausdorff implies collectionwise normal collectionwise normal not implies paracompact Hausdorff |FULL LIST, MORE INFO
Tychonoff space it is T1 and completely regular (via normal Hausdorff) (via normal Hausdorff) |FULL LIST, MORE INFO
completely regular space any point and disjoint closed subset can be separated by a continuous function to [0,1] (via normal Hausdorff) (via normal Hausdorff) Normal Hausdorff space|FULL LIST, MORE INFO