Bredon's (Compact Transformation Groups) Čech Cohomology of one point compactification

Question

In the proof of Theorem 9.4 on page 156, he writes: Let M be a connected n-dimensional orientable manifold, and X be the one point compactification of M with point A = $\{\infty\}$. If $V\subset M$ is an open n-disk then the inclusion $(X,A)\to (X,X-V)$ induces an isomorphism $$\check{\mathrm{H}}^n(D^n,S^{n-1}) = \check{\mathrm{H}}^n(X,X-V) \to \check{\mathrm{H}}^n(X,A)$$. These facts follow from Lefschetz Duality (e.g., see Spanier page 297) or by a fairly elementary direct argument. I don't have access to Spanier's book, and I really don't see the argument. Can you help? (Added 2/19/2020). Towards an answer, the first claim is that $\check{\mathrm{H}}^i = 0$ for i > n, and $\check{\mathrm{H}}^n = \mathbb{Z}$, so that this is clearly a question about Čech cohomology, and the coverings, rather than duality. So I updated the title.