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Concept
Contractible space
Summary
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.
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:
X is contractible (i.e. the identity map is null-homotopic).
X is homotopy equivalent to a one-point space.
X deformation retracts onto a point. (However, there exist contractible spaces which do not strongly deformation retract to a point.)
For any path-connected space Y, any two maps f,g: Y → X are homotopic.
For any space Y, any map f: Y → X is null-homotopic.
The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible).
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.
A topological space X is locally contractible at a point x if for every neighborhood U of x there is a neighborhood V of x contained in U such that the inclusion of V is nulhomotopic in U. A space is locally contractible if it is locally contractible at every point. This definition is occasionally referred to as the "geometric topologist's locally contractible," though is the most common usage of the term. In Hatcher's standard Algebraic Topology text, this definition is referred to as "weakly locally contractible," though that term has other uses.
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