Corollary 3

Let X be a topological space homotopic to a point. For every point $x$ of X, the group $\pi_1(X, x)$ is reduced to the identity element.

Cor. 3 applies in particular when X is the numerical space of $n$ dimensions $\mathbf{R}^n$ and more generally when X is a subset of the space $\mathbf{R}^n$ which is star-shaped (III, p. 234) with respect to one of its points.