= Compact Hausdorff quotient by a closed equivalence relation
Let $X$ be a <compact Hausdorff space> and $R\subseteq X\times X$ a closed <equivalence relation>. The <quotient space> $X/R$ is compact and Hausdorff. Indeed, the saturation $R[C]$ of any closed $C\subseteq X$ is the projection of the compact set $R\cap(C\times X)$, so is closed. Thus the quotient map is a <closed map>.
Two different equivalence classes are disjoint compact subsets of $X$ and have disjoint open neighbourhoods $U,V$. To see the latter assertion, separate each pair of points by disjoint neighbourhoods and use compactness twice to extract finite covers. Now $U'=X\setminus R[X\setminus U]$ and $V'=X\setminus R[X\setminus V]$ are open saturated subsets containing the respective classes, and lie inside $U,V$. Their quotient images are disjoint open neighbourhoods, proving the <Hausdorff condition>. Compactness follows from the continuous quotient map. This applies to finite gluing along closed boundary pieces when the generated equivalence relation is closed; that last condition must be checked.
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