Alexander duality theorem

This article is about a 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 $(M,M\setminus K)$.