TITLE: Are nets and filters useful in geometry and topology?
QUESTION [12 upvotes]: Many results in topology can be restated using the concepts of nets and ultrafilters. This seems to be of interest for set theorists, maybe even logicians. But for geometers and topologists, those who use point-set topology only as a tool in proving theorems about manifolds, varieties, schemes, homology groups, etc, can this reformulation be useful? If it is, please give examples of how it might be used.
In the case of Tychonoff's Theorem, it may provide an interesting way to prove a result in point-set topology which is useful for geometers, but even in this case, its use does not seem to shed much insight once one has obtained the technical point-set result.

REPLY [2 votes]: In formalised mathematics, Lean's mathlib library uses filters extensively when developing the basic theory of topological spaces.  I do not know exactly why that decision was taken.  However, when a non-traditional formulation turns out to be more convenient for formalisation, it is often worth asking whether that formulation has advantages in other contexts as well.