Exercise 23, § 10

Let X and Y be two locally compact spaces, f a continuous mapping from X into Y. Now prove that the following conditions are equivalent:

α) Every point $y \in f(X)$ possesses in Y a compact neighbourhood V such that $f^{-1}(V)$ is compact. β) Every point $y \in f(X)$ possesses in Y an open neighbourhood T such that the mapping $f_T : f^{-1}(T) \to T$ which coincides with f in $f^{-1}(T)$ is proper. γ) The subspace $f(X)$ of Y is locally compact, and if j is the canonical injection of $f(X)$ into Y, f factors through $X \xrightarrow{g} f(X) \xrightarrow{j} Y$, where g is proper.