Exercise 6, § 3

If $n \geqslant 0$, let $\mathfrak{F}$ be the set of all closed subsets of $\mathbf{P}_n$ endowed with the uniformity induced by that of $\mathbf{P}_n$ by the procedure of Chapter II, § 2, Exercise 6 b). Show that the set $\mathbf{P}_{n,p}$ of projective linear varieties of $p \geqslant 0$ dimensions is closed in $\mathfrak{F}$, and that the topology induced on $\mathbf{P}_{n,p}$ by the topology of $\mathfrak{F}$ is the same as that defined in no. 5, Definition 2. [To define the entourages of the uniformity of $\mathbf{P}_n$ we may consider $\mathbf{P}_n$ as a quotient space of $S_n$ (no. 1, Proposition 2) and take finite coverings of $S_n$ by balls whose radii tend to 0.]