Proposition 1

The projective space $P_n$ is Hausdorff.

We start by showing that the relation $\Delta_n$ is open (Chapter I, § 5, no. 2). Let $A$ be an open set in $\mathbf{R}_{n+1}^*$; to saturate $A$ with respect to $\Delta_n$ we have to take the union of the sets $tA$ homothetic to $A$, as $t$ runs through the set of real numbers $\neq 0$; since each of these sets is open, so is their union.

By Proposition 8 of Chapter I, § 8, no. 3, the proposition will be proved if we show that the subset $M$ of $\mathbf{R}_{n+1}^* \times \mathbf{R}_{n+1}^*$ defined by the relation $\Delta_n$ is closed. Let then $(x, y)$ be a point of $\mathbf{R}_{n+1}^* \times \mathbf{R}_{n+1}^*$ lying in the closure of $M$. If $x = (x_i)$, there is an index $i$ such that $x_i \neq 0$; hence there is a neighbourhood $V$ of $(x, y)$ such that for every point $(x', y') \in M \cap V$ the ith coordinate $x'_i$ of $x'$ is not 0. As $(x', y')$ tends to $(x, y)$ while remaining in $M$, $y'_i {x'}^{-1}_i$ tends to $t = y_i x_i^{-1}$; since $y' = (y'_i {x'}^{-1}_i) x'$, we see by passing to the limit that $y = x$, which shows that $(x, y) \in M$.