The (largest) Lebesgue number and its relative version

Abstract

In this paper we compare different definitions of the (largest) Lebesgue number of a cover U for a metric space X. We also introduce the relative version for the Lebesgue number of a covering family U for a subset A ⊆ X, and justify the relevance of introducing it by giving a corrected statement and proof of the Lemma 3.4 from [BL07], involving λ-quasi homothetic maps with coefficient R between metric spaces and the comparison of the mesh and the Lebesgue number of a covering family for a subset on both sides of the map.