Exercise 10, § 10

Let $X, Y$ be two topological spaces. A correspondence $(G, X, Y)$ between $X$ and $Y$ (Set Theory, Chapter II, § 3, no. 1) is said to be proper if the restriction of the projection $pr_2$ to the graph $G$ is a proper mapping $G \to Y$.

a) Let $f : X \to Y$ be a mapping. Show that $f$ is continuous if and only if the correspondence $\overline{f}$ between $Y$ and $X$ is proper, and that $f$ is a proper mapping if and only if the correspondences $\overline{f}$ (between $Y$ and $X$) and $f$ (between $X$ and $Y$) are proper. Hence construct an example of a non-continuous mapping $f$ such that the correspondence $f$ is proper.

b) If $(G, X, Y)$ is a proper correspondence, show that $G(A)$ is a closed subset of $Y$ whenever $A$ is a closed subset of $X$.