Corollary 1

The sphere $S_n$ is homeomorphic to the quotient space of the ball $B_n$ obtained by identifying all the points of the sphere $S_{n-1}$.

The ball $B_n$ is a regular space (Chapter I, § 8, no. 4); hence the quotient space $F$ of $B_n$ obtained by identifying all the points of $S_{n-1}$ is Hausdorff (Chapter I, § 8, no. 6, Proposition 15). Since $B_n$ is compact, so is $F$, and $F$ is therefore homeomorphic to an open ball of $n$ dimensions made compact by adjoining a point at infinity, by Alexandroff's theorem (Chapter I, § 9, no. 8, Theorem 4). The result therefore follows from Propositions 2 and 4.