Definition:Lebesgue Number

Let $$M$$ be a metric space.

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

A fixed positive real number $$\epsilon > \R$$ 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 $$M$$.