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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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    $\begingroup$ The correct spelling is "Hausdorff". $\endgroup$
    – Angelo
    Mar 10, 2013 at 16:59
  • $\begingroup$ 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. $\endgroup$
    – Benjamin Steinberg
    Mar 10, 2013 at 22:48

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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 https://math.stackexchange.com/questions/91639/x-sim-is-hausdorff-if-and-only-if-sim-is-closed-in-x-times-x).

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