Is said to be $X\approx Y$ ($X$ is homeomorphic to $Y$) iff exists a function $h: X \longrightarrow Y$ which is bijective and preserves open sets, this relationship is an equivalence relation on $Top$ (the set of all topologies).
For a set with cofinite topology (the sets are open if its complement is finite) also happens, i.e if there is a homeomorphism $h:(X,cofinite) \longrightarrow (Y,\mathcal{J})$, must be $\mathcal{J}$ the topology of cofinite?
Yes. Only finite subsets of $Y$ are closed, and any homeomorphism maps closed (finite) sets precisely to closed (finite) sets. What then can you say about the open sets in $Y$?