Proposition 8

Let $X$ be a compact space, $R$ an equivalence relation on $X$, $C$ the graph of $R$ in $X \times X$, $f$ the canonical mapping $X \to X/R$. Then the following conditions are equivalent:

a) $C$ is closed in $X \times X$.

b) $R$ is closed.

c) $f$ is proper.

d) $X/R$ is Hausdorff.

Furthermore, if these conditions are satisfied then $X/R$ is compact.

$R$ is closed if and only if $f$ is closed; hence b) implies c) by reason of Theorem 1b) of no. 2. That c) implies d) is a particular case of no. 1, Proposition 5, Corollary 2. d) implies a) for any topological space $X$ ($\S$ 8, no. 3, Proposition 8). It remains to show that a) implies b). If $F$ is a closed subset of $X$ its saturation (with respect to $R$) is $\mathrm{pr}_2(C \cap (F \times X))$; by hypothesis, $C \cap (F \times X)$ is a closed subset of the compact space $X \times X$, and is therefore compact ($\S$ 9, no. 3, Proposition 3); the result now follows from the continuity of $\mathrm{pr}_2$ ($\S$ 9, no. 4, Theorem 2, Corollary 2).

Finally it is clear that if $X/R$ is Hausdorff then it is compact ($\S$ 9, no. 4, Theorem 2).