Corollary

Let A be a commutative Banach algebra. Let $F_1$ and $F_2$ be two disjoint subsets of $\mathsf{X}(A)$, closed for the Jacobson topology. Assume that $F_2$ is weakly compact. Then there exists $u\in A$ such that $\mathscr{G}(u) = 1$ on $F_2$ and $\mathscr{G}(u) = 0$ on $F_1$.