Exercise 12, § 10

a) A correspondence $(G, X, Y)$ is proper if and only if, for every topological space $Z$ and every correspondence $(E, Z, X)$ whose graph $E$ is closed in $Z \times X$, the graph $G \circ E$ is closed in $Z \times Y$. First to show that the condition is necessary, remark that $G \circ E$ is the image of $(E \times Y) \cap (Z \times G)$ under $i_Z \times pr_2 : Z \times G \to Z \times Y$. To show sufficiency, let $\delta_Y$ be the diagonal mapping $Y \to Y \times Y$; show that, for each subset $F$ of $(Z \times Y) \times X$, the inverse image of $G \circ F$ under the mapping

is the image, under the mapping $(y, z) \mapsto (z, y)$, of the set where $\overline{F}$ is the image of $F$ under the mapping $(z, y, x) \mapsto (x, y, z)$. b) Deduce from a) that the composition of two proper correspondences is a proper correspondence. c) Let $(G, X, Y)$ be a proper correspondence. Show that if $X$ is Hausdorff then $G$ is closed in $X \times Y$.