Exercise · ZW-HEINE-BOREL-THEOREM-009 · ★★★ Advanced
The metric space \(X=(0,1)\) with the usual distance is closed (in itself) and bounded. Is it compact?
Choose one option.
A No: Heine-Borel holds in \(\mathbb{R}^n\), not in every metric space B Yes: it is closed and bounded C Yes: it is an interval D No: it is not bounded Show answer A. No: Heine-Borel holds in \(\mathbb{R}^n\), not in every metric space← Back to the question
Explanation The sequence \(1/n\) has no limit in \(X\), so \(X\) is not compact. The equivalence «closed and bounded» is special to \(\mathbb{R}^n\); in general only «compact» implies «closed and bounded».
A subset \(K\) of \(\mathbb{R}^n\) with the usual metric is compact if and only if it is closed and bounded.
\[K\subseteq\mathbb{R}^n\ \text{compact}\iff K\ \text{closed and bounded}\]
More practice problems: heine-borel theorem