De Rham cohomology of $S^n$ Can you find mistake in my computation of $H^{k}(S^{n})$.
Sphere is disjoint union of two spaces:
$$S^{n} = \mathbb{R}^{n}\sqcup\mathbb{R^{0}},$$
so
$$H^{k}(S^n) = H^{k}(\mathbb{R}^{n})\oplus H^{k}(\mathbb{R^{0}}).$$
In particular
$$H^{0}(S^{n}) = \mathbb{R}\oplus\mathbb{R}=\mathbb{R}^{2}$$
and
$$H^{k}(S^{n}) = 0,~~~k>0.$$
Where is mistake? Thanks a lot!

de Rham cohomology is not additive, unless you have a partition of the manifold $M$ by two open subsets $U$ and $V$, then the Mayer-Vietoris exact sequence implies that $H^0(M)=H^0(U)\oplus H^0(V)$. Here a point $\{N\}$ is not open in $S^n$. However, the Euler characteristic is additive and $$\chi(S^n)=\chi(\mathbb{R}^n)+\chi(\{N\})=(-1)^n+1.$$
