Exercise 5, § 1

Let $X$ be a topological space. A set $\mathscr{S}$ of subsets of $X$ is said to be saturated if it satisfies conditions $(F'_I)$ and $(F'_{II})$ of no. 2 and if the closure of every set of $\mathscr{S}$ belongs to $\mathscr{S}$.

a) Show that if $X$ is normal and if $\mathscr{S}$ is saturated and covers $X$, then the set $\mathscr{C}(X; \mathbf{R})$ is dense in the space $\tilde{\mathscr{C}}_{\mathscr{S}}(X; \mathbf{R})$ (no. 6, Theorem 2, Corollary 2) with respect to the topology of $\mathscr{S}$-convergence. In particular, $\mathscr{C}(X; \mathbf{R})$ is dense in $\mathscr{F}_s(X; \mathbf{R})$. $\mathscr{C}(X; \mathbf{R})$ is closed in $\mathscr{F}_s(X; \mathbf{R})$ only if $X$ is discrete.

b) Let $X$ be a completely regular space (Chapter IX, § 1, no. 5) and let $\mathscr{S}_1, \mathscr{S}_2$ be two saturated sets of subsets of $X$, such that $\mathscr{S}_1 \subset \mathscr{S}_2$ and $\mathscr{S}_1 \neq \mathscr{S}_2$. Show that the topology of $\mathscr{S}_1$-convergence on $\mathscr{C}(X; \mathbf{R})$ is strictly coarser than the topology of $\mathscr{S}_2$-convergence.