91Ó°ÊÓ

Prove that if \(S\) is a convex set in \(\mathbb{R}^{n}\), and \(\mathbf{x}\) is not an interior point of \(S\), then \(\mathbf{x}\) is not an interior point of \(\bar{S}\).

Show that an extreme point of a convex set must be a boundary point of the set. (Hint: Show that an interior point cannot be an extreme point.)

Let \(Q\) be the set of rational numbers. Prove that \(\bar{Q}=\mathbb{R}\) and \(\partial Q=\mathbb{R}\). What is the interior of \(\mathbb{Q} ?\)