Lebesgue number lemma

Lebesgue number lemma: For every open cover $𝒰$ 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 $𝒰$.

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