I was wondering if there are some necessary and sufficient conditions for the quotient space to be Haussdorf. I have been trying a little for a while, but I only got very restrictive sufficient conditions.

The correct spelling is "Hausdorff".
– AngeloMar 10 '13 at 16:59

You should look in Bourbaki. They have a lot on this. For a compact Hausdorff space X if R is closed in XxX then X/R is closed.
– Benjamin SteinbergMar 10 '13 at 22:48