# Definition:Lebesgue Number

## Definition

Let $M = \left({A, d}\right)$ be a metric space.

Let $\mathcal U$ be an open cover of $M$.

A fixed strictly positive real number $\epsilon \in \R_{>0}$ is called a Lebesgue number for $\mathcal U$ if and only if:

$\forall x \in A: \exists U \left({x}\right) \in \mathcal U: B_\epsilon \left({x}\right) \subseteq U \left({x}\right)$

where $B_\epsilon \left({x}\right)$ is the open $\epsilon$-ball of $x$ in $M$.

## Source of Name

This entry was named for Henri Léon Lebesgue.