Define what it means for a space to be contractible. Prove that $ℝ^n$ is contractible but $ℝ^2 ∖\{0\}$ is not. [You may assume facts about $π_1(𝕊^1)$ that you state clearly.]

A space is contractible if it is homotopy equivalent to a point. $ℝ^n$ is contractible since the homotopy $H: ℝ^n × I → ℝ^n$ defined by $H(x, t) = (1-t)x$ is a homotopy between the identity map and the constant map at 0. $ℝ^2 ∖\{0\}$ is not contractible since $ℝ^2 ∖\{0\}$ is homotopy equivalent to $𝕊^1$ and hence has fundamental group $ℤ$.