Lemma 2

Let A be a commutative Banach algebra. Let M be a subset of $\mathsf{X}(A)$. Then M is closed for the Jacobson topology if and only if, for every character $\chi \in \mathsf{X}(A)$ - M, there exists an element $x$ of A such that $\mathscr{G}(x)$ is zero on M and nonzero at $\chi$.

Let Υ(M) be the intersection of the kernels of the elements of M. The set M is closed for the Jacobson topology if and only if M = V(Υ(M)) (cf. I, p. 13 and I, p. 30). This condition is equivalent to saying that the elements $\chi$ of M are precisely the characters which vanish on Υ(M). Consequently M is closed if and only if for every character $\chi \notin M$, there exists $x\in \Upsilon (M)$ such that $\chi (x)\not= 0$. This translates into $\mathscr{G}(x)(\chi )\not= 0$ and $\mathscr{G}(x)|M = 0$.