Let $X$ be a topological space and $\sim$ be an equivalence relation defined on $X$. Let $A$ be a union of some of the equivalence classes of $X$.
So we have a well-defined map $X-A~\to~ (X/\sim) -(A/\sim)$ which takes a point in $x\in X-A$ and sends it to its equivalence class $[x]$.
This map factors through $(X-A)/\sim$ to give a bijective map $f:(X-A)/\sim~ \to~ (X/\sim) -(A/\sim)$.
Question. I am wondering if it is necessary that $f$ is a homeomorphism.
If we in addition assume that $A$ is a closed subspace of $X$, then it is clear that the map $X-A \to (X/\sim) -(A/\sim)$ is a quotient map and hence $f$ is a homeomorphism. But I am not able to prove this without the use of this additional hypothesis.
Furthermore, I have been unsuccessful at finding a counterexample.