Exercise 13, § 10

Let $X$ be a locally compact space and $Y$ any topological space. a) Show that the following four statements are equivalent: $\alpha)$ $(G, X, Y)$ is a proper correspondence between $X$ and $Y$. $\beta)$ For each Hausdorff space $X'$ which contains $X$ as a subspace, $G$ is closed in $X' \times Y$. $\gamma)$ $G$ is closed in $X \times Y$, and for each $y \in Y$ there is a neighbourhood $V$ of $y$ in $Y$ and a compact subset $K$ of $X$ such that $(X - K) \times V$ does not meet $G$. $\delta)$ There is a compact space $X'$, containing $X$ as a subspace, such that $G$ is closed in $X' \times Y$. [To prove that $\alpha) \implies \beta)$ use Exercise 12 a) and c). To show that $\delta) \implies \alpha)$, use Proposition 2 of no. 1 and Corollary 5 of Theorem 1 of no. 2.]

b) Show that if $(G, X, Y)$ is proper, then: $\zeta)$ For every compact subset $L$ of $Y$, $\overline{G}(L)$ is a compact set. Show also that if $Y$ is locally compact and if $G$ is closed in $X \times Y$ and satisfies $\zeta)$, then $(G, X, Y)$ is proper. [Use the equivalence of $\alpha)$ and $\gamma)$.] This last assertion is no longer valid when $Y$ is Hausdorff but not locally compact (Exercise 4).