Exercise 29, § 11

Let $Y$ be a locally compact, connected and locally connected space, $N$ a closed subset of $Y$ such that, for every connected nonempty open subset $V$ in $Y, V \cap (Y - N)$ are nonempty and connected. Let $X$ be a separated space, $f : X \to Y - N$ a proper local homeomorphism (exerc. 27). (a) Now prove that there exists a locally compact, connected and locally connected space $Z$, a homeomorphism $j$ of $X$ onto an everywhere dense open subset of $Z$ and a proper mapping $g$ of $Z$ into $Y$, such that $g(y)$ is finite for every $y \in Y$ and that $f = g \circ j$. (First observe that for every connected open set $W \subset Y - N$, and every connected component $C$ of $f^{-1}(W)$, one has $f(C) = W$; consequently, if $n$ is the cardinal of $f^{-1}(y)$ for every $y \in Y - N$, $^{-1}f(W)$ has at most $n$ connected components. Deduce that for every $y \in \mathbf{N}$, there are at most filters on $\mathbf{X}$ having a basis formed of connected components of the sets $^{-1}f(V)$, where V ranges over the set of connected open neighbourhoods of $y$ in $\mathbf{Y}$. Let S be the set of all these filters when $y$ ranges over $\mathbf{N}$. Take for $Z$ a sum set of $\mathbf{X}$ and S; the neighbourhoods of a point $\mathfrak{F} \in S$ are defined as follows: $\mathfrak{F}$ admits a basis formed of connected components C of sets $^{-1}f(V)$, where V ranges over the set of connected open neighbourhoods of a point $y \in \mathbf{N}$. For each of these sets C, consider the set $\tilde{C}$ of filters $F' \in S$ to which C also belongs, and the sets $\tilde{C}$ thus defined form by definition a fundamental system of neighbourhoods of $\mathfrak{F}$ in $Z$).

(b) Now prove that if $Z_1$, $g_1$ and $j_1$ are a locally compact connected and locally connected space and two mappings having the same properties as $Z$, $g$ and $j$, there exists a homeomorphism $h$ of $Z$ onto $Z_1$ such that $j_1 = h \circ j$ and $g = g_1 \circ h$. (Use exerc. 28).