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 $X$ 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 $V$.

Relation with other properties

Stronger properties

• 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