Let $X$ be compact, let $p: X \to Y$ be the quotient map, if $p$ is a closed map, then Y is Hausdorff

Question

I wonder if anyone could give me a hint, What I've done is let $x,y \in Y$, since $p$ is surjective, we have $x',y' \in X$ such that $p(x') = x $ and $p(y') = y$ but I don't know how to continue, since the singletons of $X$ are non necessarily closed. Any hint is appreciated