Contractible space

A contractible space is precisely one with the homotopy type of a point.

Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected for all n ≥ 0.

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.

Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given by Borsuk and Mazurkiewicz in their paper Sur les rétractes absolus indécomposables, C.R.. Acad.