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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.

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I'd guess that both historical reasons, as Qiaochu said, as well the fact that nets are generally [much] more complicated than sequences and such, and the mental barrier going from one to another is quite high at first. If I'd ever give an advanced course in topology (i.e. last year undergrad/first year grad students) I'd consider nets over the usual proofs. Otherwise I see no additional value (people often don't remember proofs, only the theorem). –  Asaf Karagila Jan 11 '12 at 21:42
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Do not forget that there will be those who prefer filters to nets. –  Srivatsan Jan 11 '12 at 21:47
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Nets seem nice because they superficially look like sequences, but sometimes this resemblance is misleading. In particular, the notion of a subnet is subtler than it might appear, because a subnet can have a completely different index set. For instance, a sequence is a net, but a subnet of a sequence need not be a sequence. –  Nate Eldredge Jan 11 '12 at 22:04
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@Adam, Limits: a new approach to real analysis by Alan F. Beardon. –  lhf Jan 11 '12 at 22:05
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@ Adam I learned general convergence from the classic article by Robert Bartle, "Nets and Filters In Topology", The American Mathematical Monthly Vol. 62, No. 8 (Oct., 1955), pp. 551-557. I should make you aware that Pete Clark,frequent poster here at MO, found some subtle errors in the article that he's cleaned up in some online notes of his on convergence that can found at math.uga.edu/~pete/convergence.pdf. I recommend them both most highly. –  Mathemagician1234 Jan 11 '12 at 22:17

4 Answers 4

up vote 6 down vote accepted

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.

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Munkres's Topology does cover nets, in a longish sequence of supplementary exercises at the end of Chapter 3. –  Pete L. Clark Jan 15 '12 at 20:05
    
Good observation,Pete-worth mentioning. –  Mathemagician1234 Jan 15 '12 at 20:53
    
@PeteL.Clark : Thanks! –  Adam Smith Jan 15 '12 at 22:30

I've asked that question myself of both analysts and topologists, Calvin. There are essentially 2 reasons:

1) Firstly, believe it or not, outside of research analysts - who are really the primary experts and practitioners of point-set topology in modern times - many mathematicians either have forgotten or were never taught general convergence in topological spaces. Yes, it's hard to believe, but it's true in my experience. I used to know one of the officers in the Stanford University Student Society (someone correct me if I have the name wrong, please) and we were debating the usefulness of Riemann integrals when first presenting integration in calculus. I tried to argue in addition to it's intuitive mathematical value as a constructive limit, the Riemann conception gives a good example of a net. "A what? What's a net?" This is a guy who published an original paper on non-associative algebras when he was 20. My point is that at the top graduate programs, where the goal is mainly to race students to the research frontier as quickly as possible, most mathematicians just aren't being trained with these ideas since they're not considered essential.

2) Most mathematicians who are aware of notions of generalized convergence prefer the concept of a filter over that of a net. Filters are direct set theoretic constructions. As such, they are quite a bit more elegant and in some ways simpler to work with then nets, where the notation can get quite cumbersome. I personally agree with you, it's a vastly underused tool in mathematics. Most of the properties of sequences -which anyone who's finished a strong course in calculus will know well- generalize fairly directly to nets in topological spaces and that alone makes them worth considering.

By the way, the proof of Tychonoff's Theorem using nets is due to Paul R. Chernoff, it was published in 1992, I believe.

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Chernoff gave a proof in 1992, but nets have certainly been used to prove Tychonov's theorem before. It is straightforward to rewrite the proof of Bourbaki in terms of nets. What Chernoff did was to not explicitely appeal to a universal subnet. –  Michael Greinecker Jan 11 '12 at 23:33
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@Mathemagician, while I do not entirely agree with Adam, it is frustrating when people say, "I know this because I asked so and so and he said..." in arguments. Personal correspondence is acceptable, but when you make sweeping claims about things then it starts to get bothersome. It would be like if I said that there are no graduate students who use Algebra anymore because I asked my friends at (Top School) and they said they didn't use Algebra. No one can argue against it, but no one can verify your claims either. I do not feel this argument leads to a good answer for OP's question. –  james Jan 15 '12 at 21:54
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@Adam: Set-theoretic topologists would, I think, be surprised to hear that they aren’t ‘researchers in topology’. –  Brian M. Scott Jan 17 '12 at 7:03
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@Adam: Your perception is mistaken $-$ hardly surprising, if you attend the kinds of topology conferences unlikely to attract them. You’re also excluding general topologists. You’ll find both at the annual Spring Topology Conference, in Topology Proceedings, and in Topology and its Applications. –  Brian M. Scott Jan 18 '12 at 2:51
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@Adam: You have a rather limited notion of the traditional concerns of topology. To answer your question, I probably could if I really tried, but since I’m not interested in those areas, I’d have to put in more work than it’s worth, especially in the face of such ingrained parochialism. Just one last comment: the conference and journals that I mentioned are by no means limited to set-theoretic and general topology. Some of the papers might even qualify as ‘mainstream’ in your eyes. –  Brian M. Scott Jan 18 '12 at 7:09

You should take a look at Albert Wilansky's book Topology for Analysis. In this book you will find the theory of filters and nets and you will also see how these concepts are related. It's a very good book. I prefer filters instead of nets, I think they are much more elegant, but as Wilansky says, one should not get attached. There are times when it's simpler to use nets and there are times when it's simpler to use filters. Just keep in mind that things that can be done using filters can be done using nets.

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Some thoughts on why many authors might prefer the approach via opens:

See nets are nice if you want to grasp what being and getting close means in general topological spaces sure and this way in many cases give first hints at how to obtain a proof for some theorems. That is probably also the reason why you were able to reproduce a proof for Tychonoffs theorem mostly by your own: Intuition!

However there is a big conceptual subtlety about nets for describing topology: They provide in some sense an extrinsic description, namely how do nets behave in that space. Taking a closer look on this one encounters that unfortunately that happens in most approaches: Topology is described in most approaches by relating it to order theoretic structures, some of them include the approach via open or closed sets, via neighborhoods and via filters or nets. However most prominent the approach via open or closed sets or less prominent the one via neighborhoods are better in the extend that they choose a system of open or closed sets resp. neighborhoods while the approach via filters or nets compare them among each other.

Yet nets gain a lot of attraction when it comes to the point to construct specific objects in some topological space and I guess that is where one really benefits of them. Just to name some of them think of the Riemann integral, summability in general or more advanced the dynamics in quasi local algebras.

I hope that gave you some ease why still many authors rely on the old fashoined approach by opens.

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