Covering theorems (such as Vitali covering lemma, Besicovitch covering theorem, Vitali-type covering theorem for Lebesgue measure, etc.) are crucial for analysts and considered among the most basic topics in real analysis by many people. However, some basic graduate books, like Folland's, do not cover (!) the material.
I am trying to understand what's the geometers' opinion on those theorems, whether it's useful or important for young geometers to learn them early in their careers (young analysts will have to learn about them, eventually).
What are some results, if any, in geometry or topology that require covering theorems?
Any result in any other area, aside from analysis, is also welcome.