Lebesgue number lemma

Lebesgue number lemma: Let $(X,d)$ be a metric space and assume $X$ is compact. For every open cover $\mathcal{U}$ of $X$ , there exists a number $\delta>0$ (a Lebesgue number for $\mathcal{U}$ ) such that for every $x\in X$ there is some $U\in\mathcal{U}$ with

B(x,\delta)\subseteq U,

where $B(x,\delta)$ is the open ball.

Equivalent characterizations

The ball formulation implies that every nonempty subset of $X$ with diameter less than $\delta$ is contained in a member of the cover. Conversely, if this diameter condition holds at a scale $\eta>0$ , the ball formulation holds with $\delta=\eta/2$ . Thus the two existence statements are equivalent, although the same numerical scale need not work in both directions.

Remarks

This lemma turns qualitative compactness (existence of finite subcovers) into a quantitative uniform scale, and it is a key tool in results about uniform continuity and refinements of open covers.