Let \((X,d)\) be a metric space and \(\mathcal U\) an open cover of \(X\). A real number \(\delta>0\) is a Lebesgue number for \(\mathcal U\) if every subset \(A\subseteq X\) with diameter \(\diam(A)<\delta\) is contained in some \(U\in\mathcal U\).
(Lebesgue number lemma)
Every open cover of a compact metric space has a Lebesgue number [@Mun00].