Exercise 25, § 11

Given two topological spaces $X, Y$, a mapping $f : X \to Y$ is said to be a local homeomorphism if for each $x \in X$ there is a neighbourhood $U$ of $x$ such that $f(U)$ is a neighbourhood of $f(x)$ in $Y$, and the mapping $U \to f(U)$ which coincides with $f$ on $U$ is a homeomorphism. Every local homeomorphism is a continuous open mapping. a) Suppose that $X$ is Hausdorff and $Y$ locally connected. Let $f : X \to Y$ be a local homeomorphism with the property that there is an integer $n > 0$ such that $f^{-1}(y)$ has exactly $n$ points for every $y \in Y$. Show that every $y \in Y$ has an open neighbourhood $V$ such that $f^{-1}(V)$ has $n$ components $U_i$ ($1 \leq i \leq n$) and such that the mapping $U_i \to V$ which coincides with $f$ on $U_i$ is a homeomorphism. $f$ is then a proper mapping. b) Suppose that $X$ is Hausdorff and that $Y$ is connected and locally connected. Let $f : X \to Y$ be a proper local homeomorphism. Show that $f$ satisfies the hypotheses of a). [If for each integer $n > 0$, $Y_n$ is the set of $y \in Y$ such that $f^{-1}(y)$ has at least $n$ points, show that $Y_n$ is both open and closed in $Y$]. c) Give an example of a surjective local homeomorphism $f : X \to Y$, in which $Y$ is compact, connected and locally connected, $X$ is locally compact, connected and locally connected, $f^{-1}(y)$ consists of at most two points for each $y \in Y$, but $f$ is not proper (cf. § 5, Exercise 4).