The Stacks project

Exercise 111.6.28. Let $T$ be a connected component of $\mathop{\mathrm{Spec}}(A)$. Prove that $T$ is stable under generalization. Prove that $T$ is an open subset of $\mathop{\mathrm{Spec}}(A)$ if $A$ is Noetherian. (Remark: This is wrong when $A$ is an infinite product of copies of ${\mathbf F}_2$ for example. The spectrum of this ring consists of infinitely many closed points.)