# Uses of Lebesgue's covering lemma

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?

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I recall $X$ being a compact metric space in the lemma. – Asaf Karagila Sep 19 '11 at 6:25
Oh yes, sorry, fixed. Thanks. – Lit Sep 19 '11 at 6:29

It’s not needed at all for that proof. A sequentially compact space is easily shown to be countably compact. If it’s metric, it’s totally bounded, hence separable, hence Lindelöf, and a countably compact Lindelöf space is compact. Total boundedness is easy. If for some $\epsilon>0$ $X$ has no finite $\epsilon$-net, recursively construct an infinite sequence whose points are pairwise at least $\epsilon$ apart; clearly it has no convergent subsequence. – Brian M. Scott Sep 19 '11 at 22:45