# 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 contractible space. View other variations of contractible space

## 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 .

## Formalisms

### In terms of the locally operator

*This property is obtained by applying the locally operator to the property: contractible space*

Note that the locally operator here means the existence of a basis of contractible spaces. It is a stronger condition than merely saying that every point is contained in a contractible open subset; rather, we are claiming that there are arbitrarily small contractible open subsets. The mere condition that every point is contained in a contractible open subset is much weaker.

## Relation with other properties

### Incomparable properties

- Contractible space: A contractible space need not be locally contractible; in fact, it need not even be locally connected! An example of a contractible space that is not locally connected is the comb space. Conversely, a locally contractible space need not be contractible. For instance, any manifold is locally contractible, but manifolds such as the circle are not contractible.

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

locally Euclidean space | has a basis comprising subsets homeomorphic to Euclidean space | follows from Euclidean spaces being contractible | a pair of intersecting lines is locally contractible but not locally Euclidean | |FULL LIST, MORE INFO |

manifold | locally Euclidean of fixed dimension as well as Hausdorff and second-countable | (via locally Euclidean) | (via locally Euclidean) | Locally Euclidean space|FULL LIST, MORE INFO |

CW-space | underlying topological space (up to homeomorphism) of a CW-complex | CW implies locally contractible | |FULL LIST, MORE INFO | |

polyhedron | underlying topological space (up to homeomorphism) of the geometric realization of a simplicial complex | (via CW-space) | CW-space|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

locally simply connected space | has a basis comprising subsets that are simply connected | follows from contractible implies simply connected | |FULL LIST, MORE INFO | |

semilocally weakly contractible space | |FULL LIST, MORE INFO | |||

semilocally simply connected space | |FULL LIST, MORE INFO | |||

locally path-connected space | has a basis comprising path-connected subsets | |FULL LIST, MORE INFO | ||

locally connected space | has a basis comprising connected subsets | |FULL LIST, MORE INFO |