# Paracompact Hausdorff space

From Topospaces

*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

## Contents

## 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

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 |