Alexander duality theorem

Statement
Let $$M$$ be an orientable manifold and $$K$$ a compact subset of $$M$$. Denote by $$\overline{H}^i(K)$$ the direct limit of cohomology groups for all open sets containing $$K$$. Suppose $$(M,M \setminus K)$$ is $$R$$-orientable. Choose a generator for $$H_n(M, M \setminus K)$$ (this group is a free module of rank one over the coefficient ring). Then cap product with this generator yields a map:

$$\overline{H}^i(K; R) \to H_{n-i}(M, M \setminus K ; R)$$

This map is an isomorphism.

Note that the specific isomorphism depends on the choice of orientation on the pair $$(M, M \setminus K)$$.