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!

This should be a comment but is too long.
If your reasoning were correct, we could also do the following: write $S^1$ as the "disjoint union" of two open intervals and two points (by cutting out the north and south poles, for example) Then your idea would show that $H^0(S^1)=\mathbb R^4$. And you can cut it in more pieces...
