Definition:Lebesgue Number

Definition
Let $M$ be a metric space.

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

A fixed positive real number $\epsilon \in \R: \epsilon > 0$ is called a Lebesgue number for $\mathcal U$ iff:
 * $\forall x \in M: \exists U \left({x}\right) \in \mathcal U: N_\epsilon \left({x}\right) \subseteq U \left({x}\right)$

where $N_\epsilon \left({x}\right)$ is the $\epsilon$-neighborhood of $x$ in $M$.