Exercise · ZW-LEBESGUE-NUMBER-002 · ★☆☆ Basic
If \(\delta=0.3\) is a Lebesgue number of a cover, which other value is also one with no further information?
Choose one option.
Show answer
- C. \(\delta=0.1\)
Explanation
A set of diameter less than \(0.1\) has diameter less than \(0.3\), so it fits in a member. Larger values are not guaranteed.
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
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