Exercise 2.16

Note that $E=[\sqrt{2},\sqrt{3}]\cap \mathbb{Q}$. It follows from thm. 2.23 and thm. 2.30 that $E$ is closed. Boundedness is clear.

Let $a_n$ be a sequence in $\mathbb{Q}$ that converges from above to $\sqrt{2}$ (as described in chapter 1). Let $A_n=(a_n,\sqrt{3})\cap \mathbb{Q}$. This is an open cover of $E$, but clearly has no finite subcover. $E$ is consequently not compact.

$E$ is open by thm 2.30, since $E=(\sqrt{2},\sqrt{3})\cap \mathbb{Q}$.