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

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

Also see

 * Number Smaller than Lebesgue Number is also Lebesgue Number
 * Open Cover may not have Lebesgue Number