Two topological spaces and are homotopy equivalent if there exist continuous maps and , such that the composition is homotopic to the identity on , and such that is homotopic to . Each of the maps and is called a homotopy equivalence, and is said to be a homotopy inverse to (and vice versa).
One should think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another.
Certainly any homeomorphism is a homotopy equivalence, with homotopy inverse , but the converse does not necessarily hold.
Some spaces, such as any ball , can be deformed continuously into a point. A space with this property is said to be contractible, the precise definition being that is homotopy equivalent to a point. It is a fact that a space is contractible, if and only if the identity map is null-homotopic, i.e., homotopic to a constant map.