compact subset such that A topological space is compact if and only if the map from that space to a single point is proper. If is a proper continuous map and is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then is closed. Generalization It is possible to generalize the notion of proper maps of topological spaces to locales and topoi, see . See also Citations References , esp. section C3.2 "Proper maps" , esp. p. 90 "Proper maps" and the Exercises to Section 3.6. Theory of continuous functions