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This question already has an answer here:

If a continuous function between locally compact Hausdorff spaces is proper (i.e. preimages of compact sets are compact), then it is also closed. How to prove it?

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marked as duplicate by Ayman Hourieh, user147263, Christopher, drhab, kjetil b halvorsen Jun 5 '15 at 14:31

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A much more general theorem holds.

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(cited from “General Topology” by Ryszard Engelking. )

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