Two compact $n$-manifolds $M_0, M_1$ are said to be cobordant if there is an $(n+1)$-dimensional compact manifold $M$ such that $\partial M = M_0 \sqcup M_1$.
What is the necessity of compactness here? From a naive perspective (i.e. mine) it seems as though we can just remove compactness from the definition, and just use two arbitrary $n$-manifolds $M_0, M_1$ instead.