Exercise · ZW-LEBESGUE-NUMBER-004 · ★☆☆ Basic
Why does the cover \(U_n=(1/n,1)\), \(n\ge2\), of \((0,1)\) have no Lebesgue number ?
Choose one option.
A Because the \(U_n\) are not open B Because \((0,1)\) has no diameter C Because the cover is finite D Because, for any \(\delta\), the set \((0,\delta/2)\) has diameter less than \(\delta\) and fits in no \(U_n\) Show answer D. Because, for any \(\delta\), the set \((0,\delta/2)\) has diameter less than \(\delta\) and fits in no \(U_n\)← Back to the question
Explanation Every \(U_n\) omits the points near \(0\), so no \(U_n\) contains \((0,\delta/2)\) for small \(\delta\). The \(U_n\) are open and the cover is infinite.
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 \(\mathcal{U}\).
\[\operatorname{diam}A<\delta\ \Rightarrow\ \exists\,U\in\mathcal{U}:\ A\subseteq U\]
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