Hereditarily paracompact 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

This is a variation of paracompactness. View other variations of paracompactness

Definition

Symbol-free definition

A topological space is termed hereditarily paracompact if it satisfies the following equivalent conditions:

• Every subset is paracompact in the subspace topology
• Every open subset is paracompact in the subspace topology

Formalisms

In terms of the hereditarily operator

This property is obtained by applying the hereditarily operator to the property: paracompactness

Note that for compactness-type properties in general, being hereditary on open subsets is sufficient for being hereditary on all subsets.

Relation with other properties

Stronger properties

• Metrizable space

Weaker properties

• Paracompact space
• Hereditarily metacompact space
• Hereditarily orthocompact space

Metaproperties

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

Any subspace of a hereditarily paracompact space is hereditarily paracompact; this follows from the definition.