Exercise 26, § 4

a) Let $X, Y$ be two locally compact spaces, $f$ a continuous mapping from $X$ into $Y$; suppose that for a point $y \in Y$, a connected component $C$ of $f^{-1}(y)$ is compact. Now prove that for every open neighbourhood $V$ of $y$ in $Y$, there exists an open relatively compact neighbourhood $V_1 \subset V$ of $y$, and an open relatively compact neighbourhood $U$ of $C$ in $X$, contained in $f^{-1}(V_1)$, such that the mapping $f_1 : U \to V_1$ which coincides with $f$ in $U$, is proper. (Using the corollary of II, p. 32, now prove that there exists an open relatively compact neighbourhood $W$ of $C$ contained in $f^{-1}(V)$ and such that the boundary $S$ of $W$ in $X$ does not meet $f^{-1}(y)$; take for $U$ (resp. $V_1$) the set of points of $W$ not belonging to $f^{-1}(f(S))$ (resp. the set of points of $V$ not belonging to $f(S)$).

b) Let $X, Y$ be two locally compact spaces, $f$ a continuous mapping from $X$ into $Y$, such that the connected components of all the sets $f^{-1}(y)$, where $y$ ranges over $Y$, are compact. Let $R$ be the equivalence relation in $X$ whose equivalence classes are these connected components. Now prove that the canonical mapping $X \to X/R$ is proper (use $a$), and Proposition 9 of I, p. 78).