Locally contractible space

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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

Weaker properties