Definition:Lebesgue Number

Definition
Let $M = \struct {A, d}$ be a metric space.

Let $\UU$ be an open cover of $M$.

A fixed strictly positive real number $\epsilon \in \R_{>0}$ is called a Lebesgue number for $\UU$ :
 * $\forall x \in A: \exists \map U x \in \UU: \map {B_\epsilon} x \subseteq \map U x$

where $\map {B_\epsilon} x$ 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