Proposition 5

In a projective space $\mathbf{P}_n$ ($n \geq 0$), the complement of a projective hyperplane $\mathbf{H}$ is homeomorphic to $\mathbf{R}^n$.

By making a projective linear transformation we may assume that $\mathbf{H}$ is the hyperplane whose equation is $x_0 = 0$. The set $A$ of points $x = (x_i)$ of $\mathbf{R}_{n+1}^*$ such that $x_0 \neq 0$ is open and saturated with respect to $\Delta_n$; its canonical image $C$ in $\mathbf{P}_n$, which is the complement of $\mathbf{H}$ in $\mathbf{P}_n$, is therefore homeomorphic to the quotient of $A$ by the equivalence relation $\Theta$ induced on $A$ by $\Delta_n$ (Lemma 1). Let $B$ be the hyperplane whose equation is $x_0 = 1$ in $\mathbf{R}^{n+1}$. To each point $x \in A$ let correspond the point $x_0^{-1} x$ where the line through $0$ and $x$ cuts $B$ (Fig. 4); in this way we define a continuous mapping $g$ of $A$ onto $B$ such that $g(x)$ is the only point of $B$ congruent to $x$ modulo $\Theta$. It follows that $B$ is homeomorphic to $A/\Theta$ (Lemma 2), hence to $C$; since $B$ is homeomorphic to $\mathbf{R}^n$, the proof is complete.