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:

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	quick description	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