Example

Let $X$ be a set, and let $f : X \to X'$ be a mapping of $X$ onto a topological space $X'$; topologize $X$ with the inverse image under $f$ of the topology of $X'$. Then $f$ is proper, for $f$ is closed (§ 5, no. 1, Example 3) and the inverse image of a point of $X'$ is a subspace of $X$ whose topology is the coarsest topology and is therefore quasi-compact.