Exercise · ZW-LEBESGUE-NUMBER-009 · ★★★ Advanced
What is the idea of the proof that a compact metric space \(X\) has a Lebesgue number ?
Choose one option.
A Take the smallest diameter among the members of the cover B Take \(\delta\) equal to the diameter of \(X\) C Choose the member of largest diameter D Show that \(f(x)=\max_U d(x,X\setminus U)\) is continuous and positive, and take its minimum on the compact \(X\) Show answer D. Show that \(f(x)=\max_U d(x,X\setminus U)\) is continuous and positive, and take its minimum on the compact \(X\)← Back to the question
Explanation The function \(f\) measures how deep inside some member \(x\) lies; it is continuous and positive, and on a compact set it attains a minimum \(\delta>0\). The other procedures do not guarantee that every small set fits in a member.
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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