Paracompact Hausdorff space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of 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

The second definition is the one used in algebraic topology.

Relation with other properties

Stronger properties

• Compact Hausdorff space
• Polyhedron
• CW-space: For full proof, refer: CW implies paracompact Hausdorff
• Manifold

Weaker properties

• Normal space