Talk:Proper map

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[edit] Caveat emptor

The statement

• If f : X → Y is a proper continuous map and Y is a compactly generated Hausdorff space then f is closed.

is one I proved myself. I'm not aware of a reference, but surely one exists. At any rate, it is an improvement over the prior version (which had first-countable in place of CGHaus) both in generality and correctness (the Hausdorff condition is necessary here). -- Fropuff 07:15, 6 December 2006 (UTC)

[edit] Equivalent conditions

"An equivalent, possibly more intuitive definition is as follows: we say an infinite sequence of points {pi} in a topological space X escapes to infinity if, for every compact set S ⊂ X only finitely many points pi are in S. Then a map f : X → Y is proper if and only if for every sequence of points {pi} that escapes to infinity in X, {f(pi)} escapes to infinity in Y."

I'm pretty sure this is only true in a metric space. —Preceding unsigned comment added by 202.161.2.95 (talk) 03:17, 2 June 2008 (UTC)