# Locally contractible 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 contractible space. View other variations of contractible space

## Definition

### Symbol-free definition

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

1. It has a basis of open subsets each of which is a contractible space under the subspace topology.
2. For every $x \in X$ and every open subset $V \ni x$ of $X$, there exists an open subset $U \ni x$ such that $U \subseteq V$ and $U$ is a contractible space in the subspace topology from $X$.

## 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