Two topological spaces X and Y are homotopy equivalent if there exist continuous maps f:X->Y and g:Y->X, such that the composition f°g is homotopic to the identity id_Y on Y, and such that g°f is homotopic to id_X. Each of the maps f and g is called a homotopy equivalence, and g is said to be a homotopy inverse to f (and vice versa). One should think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another. Certainly any homeomorphism f:X->Y is a homotopy equivalence, with homotopy inverse f^(-1), but the converse does not necessarily hold.