Exercise 7, § 2

Show that there exists an entourage $U$ of the uniformity of $S_1$ such that, if $f$ is an inessential mapping of a topological space $E$ into $S_1$, every continuous mapping $f' : E \to S_1$ such that $(f(x), f'(x)) \in U$ for all $x \in E$ is also inessential.