Lebesgue number lemma: For every open cover $\mathcal{U}$ of a compact metric space $X$ there exists a real number $\delta > 0$ such that every open ball in $X$ of radius $\delta$ is contained in some element of $\mathcal{U}$

Any number $\delta$ satisfying the property above is called a Lebesgue number for the covering $\mathcal{U}$ in $X$