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?
Best Answer
If I remember correctly, the basic application of this in my topology class was to prove that continuous maps from compact metric spaces were uniformly continuous.
However, the lemma is really important in algebraic topology. It is used almost everywhere where you need to cut up your domain (the interval for a path or the square for a homotopy) into sufficiently small parts. From the top of my head, we used it in the proof of Seifert-van Kampen's theorem, various results on covering spaces, probably the excision theorem for singular homology etc.