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For a manifold $M^n$, orientation is often defined as a globally consistent choice of local orientations ie. a choice of generators $\mu_x$ of $H_n(M,M-x;R)$ (this group is isomorphic to R by escision) such that every point $x \in M$ has a compact neighborhood $K$ such that there is a genreator $\mu_K \in H_n(M,M-K;R)$ that is sent by the morphism induced by inclusion to the local orientation generator $\mu_y \in H_n(M,M-y;R)$ for all $y \in K$.

My question is: is there a good reason we wish the neighborhood $K$ to be compact? Why can't it just be a plain neighborhood?

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