Does a proper map have to be continuous?

Question

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.

Answer 1

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.

Answer 2

As Lee and Mike said, yes proper map is continuous!