Exercise 14, § 4

Let $X$ be a topological space. For each function $u \in \mathscr{C}(X; \mathbf{R})$ such that $u(x) > 0$ for all $x \in X$, let $V_u$ be the set of all $f \in \mathscr{C}(X; \mathbf{R})$ such that $|f| \leq u$.

a) Show that the sets $V_u$ are neighbourhoods of $0$ in $A = \mathscr{C}(X; \mathbf{R})$ with respect to a Hausdorff topology $\mathscr{T}$ compatible with the ring structure of $A$.

b) The topology induced by $\mathscr{T}$ on $\mathscr{C}^\infty(X; \mathbf{R})$ is finer than the topology of uniform convergence in $X$. These two topologies coincide if and only if $X$ is pseudo-compact (Chapter IX, § 1, Exercise 21). If $X$ is not pseudo-compact, the topology induced by $\mathscr{T}$ on the set of constant functions is the discrete topology, so that $\mathscr{T}$ is not compatible with the $\mathbf{R}$-vector space structure of $A$.

c) Show that $A$ is a Gelfand ring with respect to the topology $\mathscr{T}$ (Chapter III, § 6, Exercise 11).