Definition 1

Let $f$ be a mapping of a topological space $X$ into a topological space $Y$. $f$ is said to be proper if $f$ is continuous and if the mapping $f \times \iota_Z : X \times Z \to Y \times Z$ is closed, for every topological space $Z$.

We shall give other characterizations of proper mappings in no. 2 and 3.

If in Definition 1 we take the space $Z$ to consist of a single point, we see that: