Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is proper if \(\inverseof{f}(K)\) is compact for every compact \(K\subseteq Y\).
Let \(f\colon X\to Y\) be a continuous map.
If \(Y\) is Hausdorff and locally compact, then \(f\) is proper if and only if \(f\) is a closed map and \(\inverseof{f}(\ts{y})\) is compact for every \(y\in Y\).
If \(X\) is Hausdorff and \(Y\) is Hausdorff and locally compact, then \(f\) is proper if and only if \(f\) is universally closed: for every topological space \(Z\), the map \(f\times\id_Z\colon X\times Z\to Y\times Z\) is a closed map.
If \(X\) is a metric space and \(Y\) is Hausdorff, then \(f\) is proper if and only if for every sequence \((x_k)_{k\geq1}\) in \(X\) such that each compact subset of \(X\) contains \(x_k\) for only finitely many \(k\), each compact subset of \(Y\) contains \(f(x_k)\) for only finitely many \(k\).
[@Lee13].