Let $(X,\tau)$ be a topological space. We assign to $(X,\tau)$ an equivalence relation $\simeq_{(X,\tau)}$ in the following way:

$x\simeq_{(X,\tau)} y$ if and only if there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$.

(Reflexivity, symmetry, and transitivity of this relation are easy to see.)

Given a non-empty set $X$ and an equivalence relation $\simeq$, is there a topology $\tau$ on $X$ such that $\simeq$ equals $\simeq_{(X,\tau)}$?