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?
im looking for an example of a set X that has two topologies T and S that has an identity function from (X,T) to (X,S) that is continuous but not homeomorphic
i already know that given any space X and two topologies T and S, (X,T) and (X,S) are never homeomorphic.
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