Definition 1. Let X and Y be two topological spaces. A map f: XY is proper if the preimage under f of any compact set is compact. Definition 2. A topological space X is said to be compactly generated if the following condition is satisfied: A subspace A is closed in X if and only if An K is closed in K for all compact subspaces KC X. Definition 3. A topological space X is locally compact if the following condition is satis- fied: For every point x E X, there is a compact subset KX that contains an (open) neighborhood of x. Let X X Y be the direct product of two topological spaces, and let p: X x Y→ X be the first-coordinate projection map. Show that p is proper if and only if Y is compact.

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