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.
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If $X$ is Hausdorff and the quotient map $X\to X/R$ is open, then $X/R$ is Hausdorff if and only if $R\subseteq X\times X$ is closed (see http://math.stackexchange.com/questions/91639/xsimishausdorffifandonlyifsimisclosedinxtimesx). 

