Proposition 10

Let X be a locally compact space, R an open Hausdorff equivalence relation on X, and let f : X \to X/R be the canonical mapping. Then X/R is locally compact, and if K' is any compact subset of X/R there is a compact subset K of X such that f(K) = K'.

The first assertion is a consequence of the facts that each x \in X has a compact neighbourhood V and that f(V) is a compact neighbourhood of f(x) (§ 5, no. 3, Proposition 5 and § 9, no. 4, Theorem 2, Corollary 1). For each y \in K' let V(y) be a compact neighbourhood of some point of f^{-1}(y) in X, so that f(V(y)) is a compact neighbourhood of y. There are a finite number of points y_i \in K' such that the f(V(y_i)) cover K'. Let K_1 be the compact set $\bigcup_i V(y_i)$ in X; we have K' \subset f(K_1); hence K = K_1 \cap f^{-1}(K') is compact (because it is closed in K_1) and f(K) = K'.