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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.