# Locally contractible 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 contractibility. View other variations of contractibility

## Contents

## Definition

### Symbol-free definition

A topological space is said to be *locally contractible* if it satisfies the following equivalent conditions:

- It has a basis of open subsets each of which is a contractible space under the subspace topology.
- For every and every open subset of , there exists an open subset such that and is a contractible space in the subspace topology from .