Contractible space


In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.

Properties

A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial.
For a topological space X the following are all equivalent:
The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one.
Furthermore, X is contractible if and only if there exists a retraction from the cone of X to X.
Every contractible space is path connected and simply connected. Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected for all n ≥ 0.

Locally contractible spaces

A topological space is locally contractible if every point has a local base of contractible neighborhoods. Contractible spaces are not necessarily locally contractible nor vice versa. For example, the comb space is contractible but not locally contractible. Locally contractible spaces are locally n-connected for all n ≥ 0. In particular, they are locally simply connected, locally path connected, and locally connected.

Examples and counterexamples