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In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a compact manifold $X$, every map f:$X$ $\rightarrow$$Y$ is proper. Here I think if f is not continuous, we can not get the claim. So I am confused if the proper map has to be continuous.

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    $\begingroup$ Yes, continuity should be part of the definition of a proper map, or at least it should be part of the hypothesis of any application of proper maps. $\endgroup$ – Lee Mosher Jul 19 '15 at 16:08
  • $\begingroup$ @LeeMosher, thank you very much! $\endgroup$ – 6666 Jul 19 '15 at 16:09
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    $\begingroup$ Likely every time they use the word 'map' they mean 'continuous map', for now and the rest of the book. $\endgroup$ – user98602 Jul 19 '15 at 16:12
  • $\begingroup$ @MikeMiller Thank you very much! $\endgroup$ – 6666 Jul 19 '15 at 16:13
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Whether to include continuity in the definition of proper maps is to some extent a matter of taste. I think the most common convention is that proper maps are not, by definition, continuous. Wikipedia follows this convention, and I follow it in my books.

The issue in the statement you quoted, "Every map from a compact manifold to another manifold is proper," is the definition of map. I don't have Guillemin & Pollack handy, but the only way this sentence can possibly be true is if the authors adopt the convention that every "map" between manifolds is automatically assumed to be continuous, or maybe even smooth. Personally, I don't like this convention, because it leads to exactly the kind of confusion that tripped up the OP.

In any case, what is true is that every continuous map from a compact space to a Hausdorff space (and hence from a compact manifold to any manifold) is proper.

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  • $\begingroup$ Thank you, that's very helpful $\endgroup$ – 6666 Jul 20 '15 at 1:39
  • $\begingroup$ Dear @Jack Lee One more question, do you assume every manifold is Hausdorff? I also saw some body talked about non-Hausdorff manifold. I am a little confused $\endgroup$ – 6666 Jul 20 '15 at 2:01
  • $\begingroup$ @Joseph: yes, part of my definition of a manifold is that it's Hausdorff and second countable. Many of the basic theorems about manifolds fail without those assumptions. $\endgroup$ – Jack Lee Jul 20 '15 at 23:13
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As Lee and Mike said, yes proper map is continuous!

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  • $\begingroup$ Could you restate what you are referring to when you say, "yes it is"? so that the answer is self-contained, otherwise I am not sure this is too helpful because it isn't grammatically an answer to the question in the title. Do you mean, "yes it does have to be continuous"? $\endgroup$ – 6005 Jul 19 '15 at 17:02

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