Proposition 3

If $n \geq 0$, the projective space $P_n$ is homeomorphic to the quotient space of the ball $B_n$ obtained by identifying each point of $S_{n-1}$ with its diametrically opposite point.

Let $H$ be the closed hemisphere of $S_n$ defined by $x_0 \leq 0$. $P_n$, which is homeomorphic to the quotient space of $S_n$ by the relation $\Delta'_n$, is also homeomorphic to the quotient of the subset $H$ of $S_n$ by the relation $\Delta''_n$ induced on $H$ by $\Delta'_n$. For every equivalence class of $\Delta'_n$ meets $H$ in at least one point, and therefore (Lemma 1) it is enough to verify that if we saturate with respect to $\Delta'_n$ an open subset $U$ of $H$ which is saturated with respect to $V''_n$, we get an open subset $V$ of $S_n$. Now if $a = (a_i) \in U$ and if $a_0 < 0$, there is a neighbourhood $W$ of $a$ in $S_n$ contained in $U$, and the union of $W$ and $-W$ is a neighbourhood of $a$ saturated with respect to $\Delta'_n$ and contained in $V$. If on the other hand $a_0 = 0$, we have $-a \in U$, and there exists $r > 0$ such that the set of points $x \in H$ which satisfy one or the other of the relations ||x - a|| < r, ||x + a|| < r is contained in U; the set of points x \in S_n which satisfy one or the other of these relations is the neighbourhood of a which is saturated with respect to $\Delta'_n$ and is contained in V.

Observe that the quotient space $H / \Delta''_n$ is obtained by identifying, in H, each point of the intersection $S_{n-1}$ of H and the hyperplane $x_0 = 0$ with its opposite point. To complete the proof it suffices to remark that the stereographic projection with vertex $e_0$ ($\S$ 2, no. 4) is a homeomorphism of H onto $B_n$ which leaves invariant the points of $S_{n-1}$.