I'm stuck on the following problem.
Let $X=S^{n}\setminus A$, where $A$ is the union of $k\geq 1$ disks $D_{k}$. Use the Mayer-Vietoris sequence to compute the de Rham cohomology $H_{\mathrm{dR}}^{*}(X)$.
For $k=1$, I see that $S^{n}\setminus D_{1}$ is diffeomorphic to $\mathbb{R}^{n}$. Since $\mathbb{R}^{n}$ is connected, we know that $H_{\mathrm{dR}}^{0}(\mathbb{R}^{n})\cong\mathbb{R}$, so $H_{\mathrm{dR}}^{0}(S^{n}\setminus D_{1})\cong\mathbb{R}$. However, I have no idea of show to handle the general case (i.e. $n\geq 0$ and $k> 1$) to find $H^{n}_{\mathrm{dR}}(S^{n}\setminus A)$. I know that I need to decompose the spaces to apply Mayer-Vietoris, but I can't seem to figure out anything past this. Any help is appreciated.
Hint: You have the $k=1$ case. By the same reasoning, for $k>1$ $X$ is diffeomorphic to $\mathbb R^n$ with $k-1$ disks removed. Use Mayer-Vietoris and induction on $k$ to compute the cohomology of such spaces.