I just finished working through a proof of Tychonoff's Theorem that uses nets (specifically, as a corollary of the fact that a net in a product space converges iff the projected nets in the components do). While I might be missing steps (I based the proofs off some optional exercises in a textbook, but the proof of the Tychonoff theorem was mostly my own), it still seemed much cleaner and certainly more than other proofs of the theorem I've seen, specifically the ones based on Zorn's Lemma/the Hausdorff Maximal Principle.
My question is why more authors don't use this method of proof. In all (two of) the topology books I've read, either the author didn't prove the theorem or used the other approach, and I'm curious why.
More generally, I'm wondering why more topology books don't talk primarily about nets and leave sequences as a special kind of net to be used in counterexamples. While there's obviously a hurdle in that you have to discuss directed sets (which are more abstract), it seems like nets would make a lot of the results about compactness, and their proofs, much cleaner.
Best Answer
There are two places in the curriculum where point-set topology is taught. The first is a course in "general topology". Here the students have (hopefully) seen the basic topology of metric spaces (eg in Rudin's small book). Books intended for this audience (such as Munkres's book, which seems to be the gold standard) often omit nets and filters. I don't know of any written explanation from eg Munkres why he made this choice, but I can speculate. The typical student here is greatly inclined to think of topological concepts in terms of sequences. Teaching them notions of generalized convergence would be misleading. Given their lack of experience, they would probably think of eg nets as "just generalized sequences", not appreciate the subtleties of things like subnets, and in the end not appreciate the strange things that can happen in arbitrary topological spaces. Moreover, they would probably not learn to think of things like continuity in terms of open sets, which is much more elegant and conceptual and also quite important in applications (eg in algebraic geometry) where you are dealing with spaces that are very much not metric spaces.
The other place where point-set topology is taught is during functional analysis courses. Here certainly many standard books (like Reed-Simon) use things like nets, and this makes sense since the students are typically more mathematically sophisticated when they take these courses.