Closedness is needed for the Hopf degree classification

The Hopf classification of The Hopf degree theorem for oriented domains assumes a nonempty compact source without boundary. For a compact manifold with boundary, prescribing values on that boundary is additional data, and a relative classification requires a separate statement for maps and homotopies of pairs (Smooth charts, atlases, and structures with boundary). No such relative classification is proved on this page.

For a noncompact source $M$ there is no proper map $M\to S^m$: the inverse image of the compact target is all of $M$, which properness would require to be compact. Thus the proper-map degree of Degree of a proper smooth map by compact-support cohomology cannot be applied to such a sphere map. For maps to other, noncompact targets the cited definition requires properness, and Degree is invariant under proper smooth homotopy requires properness of the combined homotopy. These are separate settings; this page asserts no classification there (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).