I had the following thought/conjecture: Two topological spaces are homeomorphic iff the two topologies are isomorphic. When I say that the two topologies are isomorphic, I mean that they are both monoids (the operation is union) and there is a bijective mapping f such that f(A) U f(B) = f(A U B) for all A,B in one of the topologies. Does that make sense? am I on the right track? I'll appreciate any feedback.