Exercise · ZW-HEINE-BOREL-THEOREM-010 · ★★★ Advanced
Let \(A\subseteq\mathbb{R}\) be compact and nonempty. Which statement is true?
Choose one option.
Show answer
- C. \(A\) has a maximum and a minimum
Explanation
It is bounded, so \(\sup A\) and \(\inf A\) exist; it is closed, so it contains them. \(A=[0,1]\cup[2,3]\) is neither an interval nor finite, and \(\mathbb{R}\) is the only nonempty set both open and closed in \(\mathbb{R}\), but it is not compact.
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-007 Which of these subsets of ℝ^3 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?