Cohomology of $\mathbb{S}^n$

Question

I have a corollary stating:

if X is contractible then $H_0(X) = \mathbb{R}$ and $H_n(X) = \{0\}$ for $n>0$.

But $\mathbb{S}^n$ is contractible and $H_n(\mathbb{S}^n) = \mathbb{R}$.

I must be missing something.

Thank you!

Answer 1

$\mathbb{S}^n$ is not contractible.