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Let $f \ \colon X \longrightarrow Y$ be a continuous map of topological spaces, where $X$ is Hausdorff and $Y$ is locally compact. Show that $f$ is proper (i.e. the inverse image of a compact subset of $Y$ under $f$ is a compact subset of $X$) if and only if for any topological space $Z$ the map $ X \times Z \longrightarrow Y \times Z \, , \, (x,z) \mapsto (f(x),z) $ is closed (i.e. the image of a closed subset of $X$ under $f$ is a closed subset of $Y$).