Closed equivalence relations and closed equivalence classes.

Question

Given a Compact Hausdorff space X and a closed equivalence relation on it, I am trying to show that each equivalence class is a closed subset of X. I am using the fact that the quotient space of X by this relation is necessarily Hausdorff as the relation is closed. So that by showing that each equivalence class is compact in the quotient space I'd be done, but I don't see how to show this! Any suggestions? Maybe I am missing something, could you please point me in the right direction? All help is appreciated!

Answer 1

If you already know that $X/{R}$ is Hausdorff then you're almost done: let $q: X \to X/{R}$ be the canonical quotient map sending each $x \in X$ to its class $[x]_R$ under $R$. As $X/{R}$ is Hausdorff, it is $T_1$, i.e. singletons are closed, and by definition every class $[x]_R = q^{-1}[\{q(x)\}]$, which is closed as the inverse image of a closed singleton set under the continuous $q$.