Example 4

Let E be a finite-dimensional vector space of dimension $n$ over $\mathbf{R}$ and let F be an affine subspace of E of codimension $p\geqslant 3$. The set $\mathbf{R}^p-\{0\}$ is homeomorphic to $\mathbf{R}\times \mathbf{S}_{p-1}$ (TG, VI, p. 10, cor. 2), and therefore is simply connected, since $\mathbf{R}$ and $\mathbf{S}_{p-1}$ are locally connected and simply connected (I, p. 126, prop. 4). The set E-F, homeomorphic to $\mathbf{R}^{n-p}\times$ $(\mathbf{R}^p-\{0\})$, is therefore simply connected (loc. cit.).