Homeomorphisms are equivalence relations, so what are the equivalence classes?

Question

Homeomorphisms are equivalence relations, so what are the equivalence classes for two Topological spaces $T_1, T_2$?

Intuitively it seems like we might have the following equivalence classes -

1. Functions that are homeomorphisms between both spaces
2. Functions that are not homeomorphisms between both spaces

But that's probably not correct. So can someone clarify for me what are the equivalence classes here, ie. how the spaces are partitioned?

Answer 1

Define a relation on the collection of all topological spaces by $T\sim S$ iff there exists a homeomorphism $h:T\to S$. It is easy to see that this relation is reflexive and symmetric (any space is homeomorphic to itself via the identity map, and use $h^{-1}$ is a homeomorphism $S\to T$). Transitivity will follow by simply taking compositions of homeomorphisms. Thus we have an equivalence relation. An equivalence class under this relation will by a maximal collection of topological spaces which are mutually homeomorphic.