Exercise 21, § 4

a) Let $X$ be a totally disconnected compact space, and let $A$ be a topological ring. Endow the ring $\mathscr{C}(X; A)$ with the topology of uniform convergence, which is compatible with the ring structure. Let $a$ be a left ideal of $\mathscr{C}(X; A)$ and, for each $x \in X$, let $\mathfrak{J}(x)$ be the closure in $A$ of the set of $f(x)$ for $f \in a$; $\mathfrak{J}(x)$ is a closed left ideal of $A$. Show that $\bar{a}$ is identical with the set of all $f \in \mathscr{C}(X; A)$ such that $f(x) \in \mathfrak{J}(x)$ for all $x \in X$ (note that, for each $x \in X$, the neighbourhoods of $x$ which are both open and closed form a fundamental system of neighbourhoods of $x$).

b) Suppose in addition that $A$ has a fundamental system of neighbourhoods of $o$ which are two-sided ideals. Let $B$ be a subring of $\mathscr{C}(X; A)$ which contains the constants and separates the points of $X$. Show that $B$ is dense in $\mathscr{C}(X; A)$. [For each closed subset $F$ of $X$, each $x \notin F$ and each neighbourhood $U$ of $o$ in $A$, show that there exists $f \in B$ such that $f(x) = 1$ and $f(y) \in U$ for all $y \in F$.]