AKG

Okay, and if I is the identity function from (X,T) to (X,S), V is a subset of X, what is I^-1(V)? If U is a subset X, what is I(U)? Now if I is not a homeomorphism, it has to fail to have at least one of the four properties you listed under the definition of homeomorphic. If we are trying to find an example when I is continuous, then there is in fact only one of those four properties of homeomorphism that I would fail to have. Which is it?