Exercise 3, § 4

Using Proposition 10 of Chapter I, § 3, no. 6, prove that the real projective space $P_n(\mathbf{R})$ is homeomorphic to the subspace of $P_n(\mathbf{C})$ consisting of these points which have at least one system of real homogeneous coordinates. [$R_{n+1}^*$ being considered as embedded in $C_{n+1}^*$, let $A$ be a closed set in $R_{n+1}^*$, saturated with respect to $\Delta_n(\mathbf{R})$; let $B$ be the set of all points $\zeta x$, where $x \in A$ and $\zeta \in U$; show that $\overline{B}$ is saturated with respect to $\Delta_n(\mathbf{C})$, and that $A$ is the trace of $\overline{B}$ on $R_{n+1}^*$.]