# Locally compact 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 compactness. View other variations of compactness

## Contents

## Definition

### Symbol-free definition

A topological space is termed **locally compact** if it satisfies the following equivalent conditions:

- Every point is contained in a relatively compact open neighborhood
- Every point is contained in an open set, whose closure is a compact subset
- Every point is contained in an open set, that is contained in a closed, compact subset

### Definition with symbols

A topological space is termed **locally compact** if it satisfies the following equivalent conditions:

- For every point , there exists a relatively compact open subset
- For every point , there exists an open subset , such that is compact
- For every point , there exists an open subset , and a closed compact subset of such that

## Relation with other properties

### Stronger properties

- Compact space
- Strongly locally compact space: Note that this definition coincides with the definition of locally compact if we assume the space is Hausdorff
- Locally compact Hausdorff space

### Weaker properties

## Metaproperties

### Weak hereditariness

This property of topological spaces is weakly hereditary or closed subspace-closed; in other words, any closed subset (equipped with the subspace topology) of a space with the property, also has the property.

View all weakly hereditary properties of topological spaces | View all subspace-hereditary properties of topological spaces

Any closed subspace of a locally compact space is locally compact. *For full proof, refer: Local compactness is weakly hereditary*

### Products

*This property of topological spaces is closed under taking finite products*