Exercise · ZW-LEBESGUE-NUMBER-010 · ★★★ Advanced
\(\mathbb{R}\) is not compact. Can an open cover of \(\mathbb{R}\) have a Lebesgue number?
Choose one option.
Show answer
- A. Yes: the cover \(\{(n-1,n+1)\}_{n\in\mathbb{Z}}\) admits \(\delta=1\)
Explanation
A set of diameter less than \(1\) with infimum \(a\) fits in \((n-1,n+1)\) taking \(n=\lceil a\rceil\) (if \(a\) is an integer, \(n=a\)). Compactness guarantees a Lebesgue number for every cover, but a particular cover can have one without it.
Review the concept: Lebesgue number
A Lebesgue number of an open cover \(\mathcal{U}\) of a metric space is a \(\delta>0\) such that every subset \(A\) of diameter less than \(\delta\) fits inside some member of \(\mathcal{U}\).
\[\operatorname{diam}A<\delta\ \Rightarrow\ \exists\,U\in\mathcal{U}:\ A\subseteq U\]
More practice problems: lebesgue number
- ZW-LEBESGUE-NUMBER-001 An open cover of [0,1] consists of [0,0.6) and (0.4,1]. Which of these values is a Lebesgue number?
- ZW-LEBESGUE-NUMBER-002 If δ=0.3 is a Lebesgue number of a cover, which other value is also one with no further information?
- ZW-LEBESGUE-NUMBER-003 What hypothesis on the metric space X guarantees that every open cover of X has a Lebesgue number?
- ZW-LEBESGUE-NUMBER-004 Why does the cover U_n=(1/n,1), n≥2, of (0,1) have no Lebesgue number?
- ZW-LEBESGUE-NUMBER-005 Let δ be a Lebesgue number of an open cover of a compact metric space. Which open ball, centered at…
- ZW-LEBESGUE-NUMBER-006 A cover of [0,3] consists of (-1,1.5), (1,2.5) and (2,4). What is the largest δ that is a Lebesgue…
- ZW-LEBESGUE-NUMBER-007 A cover of [0,1] consists of [0,0.7) and (0.3,1]. What is the largest δ that is a Lebesgue number?
- ZW-LEBESGUE-NUMBER-008 An open cover of [0,1] has Lebesgue number δ=0.12. What is the smallest n such that, when [0,1] is…