quality:correction, structural:expansion
Null-homotopy

A function f {\displaystyle f} is said to be null-homotopic if it is homotopic to a constant function. (The homotopy from f {\displaystyle f} to a constant function is then sometimes called a null-homotopy.) For example, a map f {\displaystyle f} from the unit circle S 1 {\displaystyle S^{1}} to any space X {\displaystyle X} is null-homotopic precisely when it can be continuously extended to a map from the unit disk D 2 {\displaystyle D^{2}} to X {\displaystyle X} that agrees with f {\displaystyle f} on the boundary.

It follows from these definitions that a space X {\displaystyle X} is contractible if and only if the identity map from X {\displaystyle X} to itself — which is always a homotopy equivalence — is null-homotopic.