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 contractibility. View other variations of contractibility

Definition

Symbol-free definition

A topological space is said to be locally contractible if given any point and any open neighbourhood of it, there exists a smaller open neighbourhood containing the point, which is contractible.

Definition with symbols

A topological space $X$ is said to be locally contractible if given any $x\in U\subset X$ with $U$ open, there exists a $V\ni x$, contained in $U$, such that $V$ is contractible (note $V$ is given the subspace topology).

Relation with other properties

Stronger properties

• Contractible space
• Manifold
• Locally Euclidean space
• CW-space

Weaker properties

• Locally simply connected space
• Semilocally weakly contractible space
• Semilocally simply connected space
• Locally path-connected space
• Locally connected space