Exercise · ZW-HEINE-BOREL-THEOREM-007 · ★★☆ Intermediate
Which of these subsets of \(\mathbb{R}^3\) is compact?
Choose one option.
Show answer
- B. \(\{|x|+|y|+|z|\le1\}\)
Explanation
The octahedron \(|x|+|y|+|z|\le1\) is closed and bounded. The set \(|x|+|y|\le1\) (a cylinder) and \(xyz=1\) (a surface) are unbounded, and \(|x|+|y|+|z|<1\) omits its boundary.
Review the concept: Heine-Borel theorem
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
- ZW-HEINE-BOREL-THEOREM-001 Identify the compact subset of ℝ:
- ZW-HEINE-BOREL-THEOREM-002 Which of these subsets of ℝ^2 is compact?
- ZW-HEINE-BOREL-THEOREM-003 Why is [0,∞) not compact in ℝ?
- ZW-HEINE-BOREL-THEOREM-004 Why is (0,1] not compact in ℝ?
- ZW-HEINE-BOREL-THEOREM-005 Among these plane figures, which one is compact?
- ZW-HEINE-BOREL-THEOREM-006 Which of these subsets of ℝ is compact?
- ZW-HEINE-BOREL-THEOREM-008 Let A=[0,1], B=(0,1), C=ℤ, D=\0\∪\1/n:n≥1\ in ℝ. How many are compact?
- ZW-HEINE-BOREL-THEOREM-009 The metric space X=(0,1) with the usual distance is closed (in itself) and bounded. Is it compact?