Proposition 3

Let $f : X \to Y$ be a continuous mapping. If $T$ is any subset of $Y$, let $f_T$ denote the mapping $\overline{f}^{-1}(T) \to T$ which agrees with $f$ on $\overline{f}^{-1}(T)$. a) If $f$ is proper, so is $f_T$. b) Let $(T(i))_{i \in I}$ be a family of subsets of $Y$ whose interiors cover $Y$, or which is a locally finite closed covering of $Y$; then if each of the mappings $f_{T(i)}$ is proper, so is $f$.

Let $Z$ be a topological space. If $T$ is any subset of $Y$, we have

if $f$ is proper, then $f \times \iota_Z$ is closed, hence so is $(f \times \iota_Z)_{T \times Z}$ [§ 5, no. 1, Proposition 2 a)], whence a) is proved. If now $(T(i))_{i \in I}$ satisfies one of the two conditions stated in b), then the covering $(T(i) \times Z)_{i \in I}$ of $Y \times Z$ has the same property; if the $f_{T(i)}$ are proper then the mappings are closed, whence $f \times \iota_Z$ is closed [§ 5, no. 1, Proposition 2 b)]. This completes the proof.