Let $ ( X , \tau )$ be a $T_2$ topological space such that every open continuous image of $X$ is $T_2$. Show that $ \tau$ is the discrete topology on $X$.
This is a question I have been thinking about and feel I am getting nowhere. The only thing I could think to try was using the theorem which says:
Lemma. If $f$ is a continuous open map of $X$ onto $Y$, then $Y$ is Hausdorff iff $ \{ (x_1, x_2) : f(x_1) = f(x_2) \}$ is a closed subset of $X \times X$.
I tried looking for some map that might let me use the lemma above to prove that any given set $A \in \tau$ is such that $ A \times A$ is closed by the lemma, and therefore $A$ is closed . I haven't had any success. Any hint or help would be appreciated, thanks!