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Let $X$ be a compact, path-connected subset of $\mathbb{R}^n$. I need a reference for the fact that the $n$-th homotopy group $\pi_n(X)$ is trivial.

EDIT: Quite embarrassing. Indeed this is false, as the comment below indicates ($\pi_3(S^2)=\mathbb{Z}$). Should have thought more before posting the question.

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    This is not true (so it's hard to give a reference). For example, $S^2$ is a compact path-connected subset of $\mathbb{R}^3$ and $\pi_3(S^2) = \mathbb{Z}$. [See also here](http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres).2011-11-05

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