# Alexander duality theorem

From Topospaces

*This article is about a duality theorem*

## Statement

Let be an orientable manifold and a compact subset of . Denote by the direct limit of cohomology groups for all open sets containing . Suppose is -orientable. Choose a generator for (this group is a free module of rank one over the coefficient ring). Then cap product with this generator yields a map:

This map is an isomorphism.

Note that the specific isomorphism depends on the choice of orientation on the pair .