Exercise · ZW-LEBESGUE-NUMBER-001 · ★☆☆ Basic
An open cover of \([0,1]\) consists of \([0,0.6)\) and \((0.4,1]\). Which of these values is a Lebesgue number?
Choose one option.
Show answer
- B. \(\delta=0.15\)
Explanation
The set \(\{0.4,\,0.6\}\) has diameter \(0.2\) and fits in no member, so \(\delta\) cannot exceed \(0.2\). Any \(\delta\le0.2\) works; of the given values only \(0.15\).
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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