It is a simple application of the Mayer-Vietoris sequence and induction to show that if $X_k$ is the Euclidean space $\mathbb{R}^n$ with $k$ points missing, then it's de Rham cohomology is given by:
$$H^0_{DR}(X_k) = \mathbb{R}$$ $$H^{n-1}_{DR}(X_k) = \mathbb{R}^k$$
What happens if an infinite amount of points is missing? If I remove a countable quantity of points on $\mathbb{R}^n$, does it yield a infinite dimensional $H_{n-1}$? Does it depend on topological properties of this set? (if the removed points are dense, for example).