Paracompact Hausdorff space

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

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
• Manifold

Weaker properties

• Normal space