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 $V$.

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

• 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