Proposition 5

Let $f : X \to X'$ and $g : X' \to X''$ be two continuous mappings. a) If $f$ and $g$ are proper, then $g \circ f$ is proper. b) If $g \circ f$ is proper and $f$ is surjective, then $g$ is proper. c) If $g \circ f$ is proper and $g$ is injective, then $f$ is proper. d) If $g \circ f$ is proper and $X'$ is Hausdorff, then $f$ is proper.

Let $Z$ be a topological space. We have

if $f$ and $g$ are proper, then $f \times \iota_Z$ and $g \times \iota_Z$ are closed; hence [§ 5, no. 1, Proposition 1 a)] $(g \circ f) \times \iota_Z$ is closed; this proves a). The proof of b) [resp. c)] runs along the same lines, using part b) [resp. c)] of Proposition 1 of § 5, no. 1, and remarking that if $f$ is surjective (resp. if $g$ is injective) then $f \times \iota_Z$ is surjective (resp. $g \times \iota_Z$ is injective). Finally, to prove d), consider the commutative diagram

where $\varphi(x) = (x, f(x))$ and $\psi(x') = (g(x'), x')$. The mapping $\varphi$ (resp. $\psi$) is a homeomorphism of $X$ (resp. $X'$) onto the graph of $f$ (resp. the reflection of the graph of $g$) (§ 4, no. 1, Proposition 1, Corollary 2). Further, since $X'$ is Hausdorff, the graph $\varphi(X)$ of $f$ is closed in $X \times X'$ (§ 8, no. 1, Proposition 2, Corollary 2). Hence (Proposition 2) $\varphi$ is proper; on the other hand Proposition 4 shows that $(g \circ f) \times i_{X'}$ is proper. By a) above and the commutativity of the diagram (1), $\psi \circ f$ is proper; but $\psi$ is injective and therefore $f$ is proper by c) above.