Proposition

Show that every convex subset of $\mathbb{R}^n$ is simply connected. Conclude that $\mathbb{R}^n$ itself is simply connected.

Solution

Let $X$ be a convex subset of $\mathbb{R}^n$. Let $p \in X$. Let $[f] \in \pi_1(X, p)$. Then the straight-line homotopy $F(s, t) = f(s) + t(c_p(s) - f(s))$ shows that $f$ is path-homotopic to $\pi_1(X, p)$. This works because $X$ is convex. Thus $\pi_1(X, p) = \{ [c_p] \}$, and thus any convex subset of $\mathbb{R}^n$ is simply connected.