\begin{definition}[Definition:Lebesgue Number]
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 $\UU$''' {{iff}}:
:$\forall x \in A: \exists \map U x \in \UU: \map {B_\epsilon} x \subseteq \map U x$
where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$ in $M$.
\end{definition}