Exercise 11, § 10

Let $(G, X, Y)$ be a proper correspondence (Exercise 10).

a) Show that if $\mathfrak{B}$ is a filter base on $X$ which consists of closed sets, then

b) For each $y \in Y$ and each neighbourhood $V$ of $\overline{G}^{-1}(y)$ in $X$, show that there is a neighbourhood $W$ of $y$ in $Y$ such that $\overline{G}^{-1}(W) \subset V$ ("continuity of the roots as functions of the parameters") (use Exercise 6 of § 9).