# Hereditarily paracompact space

From Topospaces

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

## Contents

## 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

### Weaker properties

## 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.