Proposition 8

A necessary condition for the quotient space $X/R$ to be Hausdorff is that the graph $C$ of $R$ is closed in $X \times X$. If the equivalence relation $R$ is open, this condition is also sufficient.

Let $\varphi : X \to X/R$ be the canonical mapping; then $C$ is the inverse image under $\varphi \times \varphi : X \times X \to (X/R) \times (X/R)$ of the diagonal $\Delta$ of $(X/R) \times (X/R)$. The first part of the proposition therefore follows from the continuity of $\varphi \times \varphi$ [Axiom (Hii) and Theorem 1 of § 2, no. 1]. If $R$ is open then $(X/R) \times (X/R)$ can be identified with the quotient space $(X \times X)/(R \times R)$ ($§ 5$, no. 3, corollary to Proposition 8), and $\Delta$ is then identified with the canonical image in $(X \times X)/(R \times R)$ of the set $C$, which is saturated with respect to $R \times R$. Hence $\Delta$ is closed in $X \times X$ and therefore $X$ is Hausdorff.

If $R$ is not open, there are examples where $C$ is closed but $R$ is not Hausdorff (Exercises 10 and 28).

To show that $X/R$ is Hausdorff we can also apply Proposition 5 of no. 1: $M$ and $N$ being two distinct equivalence classes of $R$ it is sufficient that there should be a continuous mapping $f$ of an open subset $A$ of $X$, saturated with respect to $R$ and containing $M$ and $N$, into a Hausdorff space $X'$, such that 1) $f$ is constant on each equivalence class mod $R$ contained in $A$, 2) $f$ takes distinct values on $M$ and $N$. For since $A/R_A$ can be identified with an open subset of $X/R$ ($§ 3$, no. 6, Proposition 10, Corollary 1), we can apply Proposition 5 of no. 1 to the mapping $g : A/R_A \to X'$ induced by $f$, since $g$ is continuous ($§ 3$, no. 4, Proposition 6).

In particular: