quotient topology in compact hausdorff space

Question

Let X is compact hausdorff space and A is a subset of X. Show that X/A is Hausdorff ,when A is closed. I have no idea how to proof it. Can you give me a clue?

Answer 1

Hint: Compact Hausdorff space is regular (even normal).

Answer 2

Split this into cases. Consider the case where $[x],[y] \in [X \setminus A]$, and then the case where $[x] \in [X \setminus A]$, $[y] \in [A]$.