Corollary 4

Let $X$ and $Y$ be two topological spaces, $f : X \to Y$ a continuous mapping, $R$ the equivalence relation $f(x) = f(y)$ on $X$, and

the canonical decomposition of $f$. Then for $f$ to be proper it is necessary and sufficient that $p$ is proper, $h$ a homeomorphism and $f(X)$ a closed subset of $Y$.

The conditions are sufficient by virtue of Proposition 5 a) and Proposition 2. Conversely, if $f$ is proper, then $f$ is closed; hence $f(X)$ is closed in $Y$ and $h$ is a homeomorphism ($§ 5$, no. 2, Proposition 3); also $h \circ p$ is proper by Proposition 5 c); hence $p = h^{-1} \circ (h \circ p)$ is proper by Proposition 5 a).