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 $$M$$.