Exercise 28, § 11

Let $Y, Z$ be two locally compact spaces, $h$ a proper mapping of $Y$ into $Z$ such that, for every $z \in Z, h^{-1}(z)$ is finite. Let $X$ be a locally connected space, $M$ a closed subset of $X$ such that, for every nonempty connected open $U$ in $X, U \cap (X - M)$ is nonempty and connected. Suppose that there exists a continuous mapping $f$ of $X - M$ into $Y$ such that the mapping $g = h \circ f$ of $X - M$ into $Z$ can be extended by continuity to a mapping $\bar{g}$ of $X$ into $Z$. Prove that under these conditions $f$ can be extended by continuity to a mapping $\bar{f}$ of $X$ without $Y$. (Consider a point $x \in M$ and its image $z = \bar{g}(x)$, as well as the filter $\mathfrak{F}$ on $X - M$ traced on $X - M$ by the filter of neighbourhoods of $x$ in $X$. Prove that the set of points adherent to $f(\mathfrak{F})$ in $Y$ is finite and nonempty and use the fact that $\mathfrak{F}$ admits a basis formed by connected sets).