Exercise 24, § 10

Let $Z, Y$ be two locally compact spaces, $f$ a proper mapping from $Z$ into $Y$, $F$ a closed subspace of $Z$ such that $X = Z - F$ is an everywhere dense open set in $Z$ and such that the restriction of $f$ to $F$ is injective (so that this restriction is a homeomorphism of $F$ onto a closed subspace $Y_0 = f(F)$ of $Y$). Now prove that $Y_0$ is the set of the $y \in Y$ such that $X \cap f^{-1}(V)$ is not compact for any neighbourhood V of $y$ in $Y$. Moreover, the topology of $Z$ is generated by the set formed by the open subsets of $X$, the complements in $Z$ of the compact subsets of $X$ and the inverse images $f^{-1}(U)$, where U runs through the set of open subsets of $Y$. (To prove that the topology generated by this set is finer than the given topology on $Z$, one may use Proposition 1 of I, p. 60).