Exercise 12, § 1

Let $X$ be a topological space and let $A$ be a topological ring (Chapter III, § 6, no. 3). The set $\mathscr{C}(X; A)$ of continuous mappings of $X$ into $A$ is a subring of the ring $A^X$. Let $\mathscr{S}$ be a set of subsets of $X$.

a) Suppose that, for each $M \in \mathscr{S}$ and each $u \in \mathscr{C}(X; A)$, the set $u(M)$ is bounded (Chapter III, § 6, Exercise 12). Show that the topology of $\mathscr{S}$-convergence is compatible with the ring structure of $\mathscr{C}(X; A)$. Consider the case of pointwise convergence and the case of compact convergence.

b) Let $X$ be a locally compact $\sigma$-compact space which is not compact (Chapter I, § 9, no. 9). Show that the topology of uniform convergence is not compatible with the ring structure of $\mathscr{C}(X; \mathbf{R})$.