Exercise 25, § 10

Let $X, Y$ be two locally compact spaces, and let $g$ be a continuous mapping from $X$ into $Y$. Prove that there exist a locally compact space $Z$, a proper mapping $f$ from $Z$ into $Y$ and a homeomorphism $j$ of $X$ onto an open everywhere dense subset of $Z$ such that $g = f \circ j$ and that the restriction of $f$ to $Z - j(X)$ is injective. If moreover $Z_1, f_1$ and $j_1$ are a locally compact space and two mappings having the same properties as $Z$, $f$ and $j$, there exists one and only one homeomorphism $h$ of $Z$ onto $Z_1$ such that $j_1 = h \circ j$ and $f = f_1 \circ h$. (Use Ex. 24 to define $Z$ and $f$).