Exercise 2, § 1

Let $X$ be a set, let $Y$ be a Hausdorff uniform space containing more than one point, and let $\mathscr{S}_1, \mathscr{S}_2$ be two sets of subsets of $X$ which satisfy conditions $(F'_I), (F'_{II})$ of no. 2. Show that if $\mathscr{S}_1 \subset \mathscr{S}_2$ and $\mathscr{S}_1 \neq \mathscr{S}_2$, then the uniformity of $\mathscr{S}_1$-convergence is strictly coarser than the uniformity of $\mathscr{S}_2$-convergence. In particular:

(i) If $X$ is a non-compact Hausdorff topological space, the uniformity of compact convergence is strictly coarser than the uniformity of uniform convergence.

(ii) If $X$ is a Hausdorff topological space which has infinite compact subsets (cf. Chapter I, § 9, Exercise 4), then the uniformity of pointwise convergence is strictly coarser than the uniformity of compact convergence.