Inmathematics, atopological spaceX iscontractible if theidentity map onX is null-homotopic, i.e. if it ishomotopic to someconstant map.[1][2] Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.
A contractible space is precisely one with thehomotopy type of a point. It follows that all thehomotopy groups of a contractible space aretrivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, sincesingular homology is a homotopy invariant, thereduced homology groups of a contractible space are all trivial.
For a nonempty topological spaceX the following are all equivalent:
Thecone on a spaceX 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 contractibleif and only if there exists aretraction from the cone ofX toX.
Every contractible space ispath connected andsimply connected. Moreover, since all the higher homotopy groups vanish, every contractible space isn-connected for alln ≥ 0.
A topological spaceX islocally contractible at a pointx if for everyneighborhoodU ofx there is a neighborhoodV ofx contained inU such that the inclusion ofV is nulhomotopic inU. A space islocally 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. InHatcher's standard Algebraic Topology text, this definition is referred to as "weakly locally contractible," though that term has other uses.
If every point has alocal base of contractible neighborhoods, then we say thatX isstrongly locally contractible. Contractible spaces are not necessarily locally contractible nor vice versa. For example, thecomb space is contractible but not locally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locallyn-connected for alln ≥ 0. In particular, they arelocally simply connected,locally path connected, andlocally connected. The circle is (strongly) locally contractible but not contractible.
Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given byBorsuk andMazurkiewicz in their paperSur les rétractes absolus indécomposables, C.R.. Acad. Sci. Paris 199 (1934), 110-112).
There is some disagreement about which definition is the "standard" definition of local contractibility[citation needed]; the first definition is more commonly used in geometric topology, especially historically, whereas the second definition fits better with the typical usage of the term "local" with respect to topological properties. Care should always be taken regarding the definitions when interpreting results about these properties.