Contractibility

Contractibility is a basic notion in topology describing spaces that can be continuously shrunk to a point. A space X is contractible if the identity map id_X is homotopic to a constant map. Equivalently, there exists a point x0 in X and a continuous function H: X × I → X such that H(x,0) = x for all x ∈ X and H(x,1) = x0 for all x ∈ X. In particular, X has the homotopy type of a point.

Consequences and related facts: If X is nonempty and contractible, it is path-connected. All higher homotopy groups vanish: π_n(X) = 0 for all n ≥ 1, and all reduced homology groups Ĥ_n(X) vanish. Thus X and a single point have the same homotopy type. Contractibility implies simple connectivity, but the converse is false in general; for example, spheres S^n (n ≥ 2) are simply connected but not contractible. If X is contractible and Y is contractible, then the product X × Y is contractible.

Examples: Euclidean space R^n is contractible, as are convex subsets of R^n and open balls. The cone over any space is contractible (including cones over spheres). Star-shaped subsets of R^n, contractible CW-complexes, and contractible manifolds (including infinite-dimensional Hilbert spaces) are typical examples.

Notes on relationships: A space is contractible if and only if it is homotopy equivalent to a point. This aligns with the intuition that a contractible space can be continuously deformed to a single point without tearing or gluing.

See also: homotopy, homotopy equivalence, fundamental group, homology, topological spaces.