Consider the metric space \((\mathbb{Q},d)\) where \(\mathbb{Q}\) denotes the rational numbers and \(d(x,y)=|x-y|\). Let \(E=\{x\in\mathbb{Q}:x>0,\,2<x^2<3\}\). Is \(E\) closed and bounded in \(\mathbb{Q}\)? Is \(E\) compact in \(\mathbb{Q}\)?
Solution
1. \(E = \QQ \cap (\sqrt2, \sqrt3)\), in particular \(E\) is bounded.
\(E = \{q \in \QQ : q > 0,\ 2 < q^2 < 3\} = \{q \in \QQ : \sqrt2 < q < \sqrt3\}\), and \(\sqrt3 < 2\), so \(E \subseteq (0,2)\) is bounded.
2. \(E\) is closed in \(\QQ\).
\(E = \QQ \cap (\sqrt2, \sqrt3) = \QQ \cap [\sqrt2, \sqrt3]\), since \(\sqrt2, \sqrt3 \notin \QQ\).
3. \(E\) is NOT compact.
choose rationals \(q_n\) with \(\sqrt2 < q_n < \sqrt2 + 1/n\) (possible by density of \(\QQ\)); then \(q_n \in E\) for all \(n\), and \(q_n \to \sqrt2\) in \(\RR\).
4. Q.E.D.