Proposition 2

The projective space $P_n$ is compact and connected, and is homeomorphic to the quotient of the sphere $S_n$ by the equivalence relation induced on the sphere by $\Delta_n$.

Let $\Delta'_n$ be the equivalence relation induced on $S_n$ by $\Delta_n$ (the equivalence classes of $\Delta'_n$ are pairs of diametrically opposite points of $S_n$). The mapping $x \to x/||x||$ of $\mathbf{R}_{n+1}^*$ onto $S_n$ is continuous, hence (Lemma 2) $P_n$ is homeomorphic to $S_n / \Delta'_n$. Since $S_n$ is compact and connected, every Hausdorff quotient space of $S_n$ is also compact and connected (Chapter I, § 9, no. 4, Theorem 2, Corollary 1; § 11, no. 3, Proposition 7).