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I have encountered a characterization of compact space, but I do not know how to prove it.

$V$ is a topological space which satisfies that for any topological space $W$, the projection $V\times W\rightarrow W$ is a closed map, then $V$ is compact.

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    Your question is equivalent to the statement that the function from $V$ to a one point space is a [proper map](http://en.wikipedia.org/wiki/Proper_map). Read more about proper maps in Bourbaki's general topology, for example.2012-11-01

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