Proposition 4

If $n > 1$, the Euclidean sphere $S_{n-1}$ is homeomorphic to the space $\mathbf{R}^{n-1}$ made compact by adjoining a "point at infinity"(Chapter I, § 9, no. 8, Theorem 4).

For the stereographic projection defines a homeomorphism of the complement of a point in $S_{n-1}$ onto a hyperplane of $\mathbf{R}^n$, which is homeomorphic to $\mathbf{R}^{n-1}$.