Suppose $n=i+j,$ with $n, i,j$ positive integers. Let $I^k$ denote the $k$-dimensional unit square.
It is claimed (in Hatcher's Algebraic Topology text) that $H^i(\mathbb{R}^n, \mathbb{R}^n \setminus \mathbb{R}^j)$ is isomorphic to $H^i(I^i, \partial I^i).$ Why is this true? My guess is that I need to use excision, but I don't see how.
There is a (linear) deformation retract of pairs $(\mathbb{R}^n, \mathbb{R}^n \setminus \mathbb{R}^j) \to (\mathbb{R}^i, \mathbb{R}^i \setminus 0)$. Now use excision to change to $(I^i, I^i \setminus 0)$. Finally the inclusion $(I^i, \partial I^i) \to (I^i, I^i \setminus 0)$ is a homotopy equivalence $\partial I^i \to I^i \setminus 0$, so induces isomphisms on relative (co)homology by the five lemma.