Exercise 14, § 4

Let $X$ be a topological space. For every 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 functions $f \in \mathscr{C}(X; \mathbf{R})$ such that $|f| \leq u$. a) Now prove that the $V_u$ are the neighbourhoods of 0 in $A = \mathscr{C}(X; \mathbf{R})$ for a separated topology $\mathscr{T}$ compatible with the ring structure of $A$. b) The topology induced by $\mathscr{T}$ on $\mathscr{C}^0(X; \mathbf{R})$ is finer than the topology of uniform convergence in $X$; in order that these two topologies be identical, it is necessary and sufficient that $X$ be weierstrassian (IX, p. 88, Exer. 20). If $X$ is not weierstrassian, 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 vector space structure (over $\mathbf{R}$) of $A$. c) Now prove that for the topology $\mathscr{T}$, $A$ is a Gelfand ring (III, p. 81, Exer. 11).