At page 353 of Bredon's Topology and Geometry, there is stated the Alexander Duality as Corollary $8.7$.
I don't understand where does the upper row come from and why is it exact.
I thought it was reduced cohomology (which I think it's the most natural choice, but according to me the direction of map are wrong, because reduced cohomology is defined as the kernel of the dual of the inclusion map $\ast \hookrightarrow X$. The fact that I can substitute $H^0(S^n)$ with $H^n(\ast)$ should comes from the fact that I can fit in a commutative diagram.