Exercise 19, § 10

Let X be a locally compact $\sigma$-compact space, R an equivalence relation on X whose graph is closed in $X \times X$. Show that $X/R$ is Hausdorff. [Let $(U_n)$ be a covering of X by relatively compact open sets such that $\overline{U}_n \subset U_{n+1}$. Let A, B be two disjoint closed sets which are saturated with respect to R. Define recursively two increasing sequences $(V_n), (W_n)$ of closed sets saturated with respect to R, such that $V_n \cap W_n = \varnothing$ and such that $V_n \cap \overline{U}_n$ (resp. $W_n \cap \overline{U}_n$) is a neighbourhood of $A \cap \overline{U}_n$ (resp. $B \cap \overline{U}_n$) in the compact subspace $\overline{U}_n$. Use Proposition 8 of no. 3 and Exercise 16, as well as Proposition 2 of § 9, no. 2]. (*)