Proposition 2

Let $f : X \to Y$ be a continuous injection. Then the following three statements are equivalent: a) $f$ is proper. b) $f$ is closed. c) $f$ is a homeomorphism of $X$ onto a closed subset of $Y$.

We have just seen that a) implies b). Since the equivalence relation $f(x) = f(x')$ is the equality relation, the quotient space of $X$ with respect to this relation can be identified with $X$; hence b) implies c) by reason of § 5, no. 2, Proposition 3. Finally, if c) is satisfied then $f \times \iota_Z$ is a homeomorphism of $X \times Z$ onto a closed subspace of $Y \times Z$ and is therefore a closed mapping; hence c) implies a).