Exercise 20, § 10

a) Let X be a non-empty compact space and R a Hausdorff equivalence relation on X. Show that $X/R$ is closed in the space $\mathfrak{P}_0(X)$ with the topology $\mathfrak{T}_\Theta$ (§ 8, Exercise 12) if and only if the relation R is open. [Use Exercise 13 of § 9 and Exercise 16 of § 3 to show that if $X/R$ is closed in $\mathfrak{P}_0(X)$ in the topology $\mathfrak{T}_\Theta$, then the quotient topology on $X/R$ coincides with that induced by $\mathfrak{T}_\Theta$, and then apply Exercise 7 of § 5. Conversely, if R is open, use Exercise 29 of § 8 to show that $X/R$ is closed with respect to $\mathfrak{T}_\Theta$].

b) Let X be the locally compact subspace of the real plane $\mathbf{R}^2$ consisting of the points $(1/n, y)$ where n is an integer > 0 and $-1 \leq y \leq 1$, and the points $(0, y)$ where either $-1 \leq y < 0$ or $0 < y \leq 1$ or $y = 2$. Let $p$ be the restriction of $pr_1$ to X, and R the equivalence relation associated with $p$. Show that R is neither open nor closed, but that $X/R$ is compact and closed in $\mathfrak{P}_0(X)$ in the topology $\mathfrak{T}_\Theta$. \

(*) If the locally compact space X is not $\sigma$-compact, it can happen that the equivalence relation R on X is closed [and a fortiori has a closed graph (§ 8, no. 6, Proposition 14)] but that $X/R$ is not Hausdorff [cf. Chapter IX, § 4, Exercises 9 b) and 14].