This Wikipedia page gives a description of the concept of a net.
Here, I am going to ask a simple question: what topics in mathematics use nets frequently, and what are some "big" theorems proven by arguments about nets?
After some search with the word "net", I have found a lot of description of its "basic" properties, but I find it difficult to discover some more "advanced" or "developed" applications of this concept.
The word "net" means a lot of different things, which makes this search extremely difficult. Could anyone provide some reference?
Nets are a tool that can be handy in proving some elementary facts about some topological spaces. They “look like” sequences, and so they seem to be most commonly used in Analysis areas. You get that continuity is equivalent to preserving net convergence, compactness can be characterised by cluster points of nets (like sequential compactness in metric spaces), it gives a didactically attractive way to prove Tychonoff’s theorem etc. (If a teacher doesn’t want to go into filters, e.g.).
They’re not studied as much as a construct in their own right, just as an alternative to filters to study general convergence (filters are much more popular, BTW, especially in areas touching more on set theory; more general and an internal tool in set theory as well (forcing etc, ultraproducts in logic etc). They occur in some proofs, but can often be avoided (but some proofs look natural done in terms of nets). There are hardly any deep theorems or hard results that require them. All in all quite a marginal topic IMO.
Schuchter (handbook of foundations of analysis or some such title) has a good modern treatment, Kelley’s general topology is the classic introduction. Engelking’s general topology does a decent job too. In Munkres they’re just a “footnote” exercise.