Exercise 2, § 3

Identify the projective space $P_n$ with the quotient space of $B_n$ defined in no. 1, Proposition 3, and let $\varphi$ denote the canonical mapping of $B_n$ onto $P_n$. A continuous mapping $f$ of a topological space E into $P_n$ is said to be inessential if there exists a continuous mapping $g$ of E into $B_n$ such that $f = \varphi \circ g$, and essential in the converse case (cf. § 2, Exercise 6). Show that there exists an entourage U of the uniformity of $P_n$ such that, if $f$ is an inessential mapping of a topological space E into $P_n$, every continuous mapping $f'$ of E into $P_n$ such that $(f(x), f'(x)) \in U$ for all $x \in E$ is also inessential.