Connectedness is needed for a single degree invariant

Connectedness is load-bearing in both Hopf classifications of this page, The Hopf degree theorem for oriented domains and The Hopf mod-two degree theorem for nonorientable domains (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets). If a closed oriented smooth $m$-manifold $M$ is not connected, each connected component is itself a closed connected oriented $m$-manifold, a smooth map $M\to S^m$ has one signed degree contribution on each component, and the total degree is the sum of these contributions; but a homotopy of maps of $M$ restricts to a homotopy on each component, so the componentwise degrees are invariants that a single integer need not capture. For a disconnected source with nonorientable components, the same invariance remark applies to their mod-two degrees. Orientable components retain their integer degrees; calling the whole source nonorientable does not make every component nonorientable. The classification theorems of this page therefore assume connectedness, and the counterexample on the companion page exhibits two maps of a disconnected closed oriented domain whose total degrees agree while the maps are not homotopic. No claim is made here that every disconnected domain admits a finer classification by the vector of componentwise degrees and their homotopy types; the recorded fact is only the failure of the total degree as a single complete invariant, witnessed by the companion counterexample (Degree of a proper smooth map by compact-support cohomology, Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints).