TITLE: Uses of Lebesgue's covering lemma
QUESTION [6 upvotes]: Consider Lebesgue's covering lemma in the following form:

Let $(X,d)$ be a compact metric space and let $\{U_i\}_{i\in I}$ be an open cover of $X$. Then there exists $\delta>0 $ such that each subset $Y$ of $X$ of diameter less than or equal to $\delta$ lies within some $U_i$. 

What are possibly the important and striking uses of this lemma named after a famous mathematician? I have seen only one use and that was in the derivation of the fundamental group of the circle, using $\mathbb R$ as the universal cover. However I can't imagine that this is the only one, especially because in the more general setting of covering spaces, it is possible to do without this lemma. Is this lemma more fundamentally important befitting its name, and if so, what are some uses to convince myself?

REPLY [3 votes]: This lemma is the key ingredient in the proof that, for metrizable spaces, topological compactness is the same as sequential compactness. Cfr. Munkres Topology, chapter 3, theorem 28.2.