The (largest) Lebesgue number and its relative version

In this paper we compare different definitions of the (largest) Lebesgue number of a cover $\mathcal{U}$ for a metric space $X$. We also introduce the relative version for the Lebesgue number of a covering family $\mathcal{U}$ for a subset $A\subseteq X$, and justify the relevance of introducing it by giving a corrected statement and proof of the Lemma 3.4 from S. Buyalo - N. Lebedeva paper"Dimensions of locally and asymptotically self-similar spaces", involving $\lambda$-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.