Exercise 7, § 1

Let $A$ be a commutative algebra over a commutative field K$,x\in A$ and $\alpha \in K$. The set of $\chi \in \mathsf{X}(A)$ such that $(\mathscr{G}x)(\chi ) =\alpha$ is closed for the Jacobson topology. (Reduce to the case where A possesses a unit element, then to the case where $\alpha = 0$.)