In the text Topology by James Dugundji there are a lot of impressively short proofs (Ex. Tychonoff. Chap. XI Theorem 1.4(4)) using filterbases that later, when I go to class, take an eternity to prove using nets (in class we are using nets). I've never found a proof become simpler once one uses nets instead of filterbases which leads to my first question:
Why would anyone ever use nets instead of filterbases?
My second question has to do with popularity of the concepts:
Which is used more often; filterbases or nets?
Again, the text I use uses filterbases, in class we use nets, if you look for a proof of Tychonoff's theorem, it seems that most use nets (even though, with filterbases, its a three-liner), and there is no tag on MSE for filterbases. Currently, I use filterbases exclusively unless it is something I have to hand to my professor (say, homework. This is due to the fact that we haven't touched on filterbases in class). Should I switch to nets? should I keep using only filterbases? should I do my best to learn both?
Best Answer
Historically the concept of net appeared earlier than the concept of filterbase. Quotation from Wikipedia:
Both concepts are equally well adequate to introduce the concept of convergence in topological spaces.
The concept of net is an obvious generalization of the concept of sequence (which is one of the most basic concepts in calculus), and in my opinion it is more intuitive than that of filterbase. Nets which are more general than sequences occur quite naturally in calculus, for example when the Riemann integral is introduced. See my answer to What is the motivation for sequences to be defined on natural numbers?
Also Dugundji starts with sequences ands nets in his chapter "Convergence". At the end of section X.1 he says
That is, the concept of a filterbase is an abstraction of that of a net. It is a very useful abstraction as you have outlined in your question, and frequently it allows to give short and elegant proofs. Your question
suggests that you think that filterbases are superior to nets. This has already been discussed in this forum. See for example
Both concepts are equivalent. To some extent it is a matter of taste whether one works with filterbases or nets. You should know both concepts which gives you the flexibility to use more than one tool. And be aware the nets do occur in calculus, replacing them by filterbases would be hard work and lead to a less intuitive approach.
This is difficult to answer. But a search in this forum gives the following results:
This suggest that nets are still more popular than filterbases. Whether this is true also in the literature is not known to me. But a search in Google Scholar gives at least an indication: