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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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    Historical inertia.2012-01-11
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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).2012-01-11
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    Do not forget that there will be those who prefer filters to nets.2012-01-11
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    Any references for introductions to filters and nets?2012-01-11
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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.2012-01-11
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    @Adam, *Limits: a new approach to real analysis* by Alan F. Beardon.2012-01-11
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    @Adam: I seem to remember that Völker Runde's *A Taste of Topology* was quite fond of nets. I'm sure he proved Tychonoff's theorem that way.2012-01-11
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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 http://math.uga.edu/~pete/convergence.pdf. I recommend them both most highly.2012-01-11
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    Bredon's Topology and Geometry has a nice account of Tychonoff and the proof is given using nets. When you mention Zorn's lemma, it's actually known that Tychonoff is equivalent to Zorn's lemma. Thus you can't prove it without using it. When working with nets you hide Zorn's lemma into some lemmas involving universal nets. I don't remember the complete details, but you can look it up in Bredon's book.2012-01-11
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    Not really. You need only a weaker notion of choice (BPI) for the convergence arguments. For Hausdorff spaces, where limits are unique, this is enough. For non-Hausdorff spaces, you have to pick a limit for each coordinate and this is where the full axiom of choice is needed.2012-01-11
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    Willard’s *General Topology* uses ultranets to prove the Tikhonov theorem. Dugundji’s *Topology* uses essentially the same short argument via filterbases.2012-01-12

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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 3.2012-01-15
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    Good observation,Pete-worth mentioning.2012-01-15
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    @PeteL.Clark : Thanks!2012-01-15
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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.2012-01-11
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    @Micheal Well,I'm not sure how well known the proof by Bourbaki is to the current generation of American mathematicians and mathematics students.But you're right in what the key difference is in Chernoff's proof and that's really what gives his version it's directness and relative simplicity.2012-01-11
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    I'm not voting on this answer in either direction but I think your point 2) gives a genuinely false impression and I don't buy 1) either. Anyone who has a decent education in topology *will* be aware of both nets and filters and it is certainly not true that filters are preferred over nets by "Most mathematicians who **are** aware of notions of generalized convergence" (I wonder whom among 'pure' mathematicians this sentence actually excludes). It depends very much on the situation which of the two is more suitable.2012-01-12
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    For what it's worth: Googling this site with `net topology site:math.stackexchange.com` and `filter topology site:math.stackexchange.com` gives me about 400 results for the former and 500 for the latter. Similarly, simply searching for `net limit topology` and `filter limit topology` gives about 5 million hits each. This doesn't look like there is a genuine preference for either notion.2012-01-12
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    @t.b. I was merely speaking from my own experience. And in case you haven't noticed, with the exception of some hardcore analysts and some other creative souls in other fields, point set topology-sadly-has fallen far out of favor lately. They don't even offer a course in it at many universities now.2012-01-12
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    As if "offered course" = "people use it". Maybe it's just easy enough to pick up the things you need along the way while focusing on other topics? I myself never had a point set topology course and I never missed it. Finally if you're speaking from your own perspective then you should maybe indicate that this is the case.2012-01-12
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    @Mathemagician1234 : The only place you've ever been at is Queen's college, and there only as an undergrad and unsuccessful masters student. I don't think you're in a position to say what "most" mathematicians know or think. Point-set topology is taught at the vast majority of places (in some form or another -- the course isn't always called "point-set topology"), and I would be surprised if any working mathematician was not aware of the rudiments of it. And I just went and quizzed the grad students hanging out in the tea room at my university, and they all knew what nets and filters were.2012-01-12
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    I have no idea why this is the accepted answer. First, your primary first claim is this: many mathematicians just "forgot" or "were never taught" general convergence. Your argument is: some very good mathematician you know didn't know what they were. This is an extremely poor argument. I'd also like to note that your stab at top graduate programs "racing" students to the research frontier just comes off as rude and somewhat insulting to those students who studied hard to get into those universities.2012-01-15
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    Additionally, I believe the second statement to be misleading. This is almost like saying, "the people who know what a topology is prefer to work with open sets rather than closed sets." I'm not trying to criticize this comment for the sake of insulting you, but in case someone later comes to look at this question, I do not want them thinking this answer represents the general thoughts of the community.2012-01-15
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    @james It's the accepted answer because it looked legitimate to me (someone with almost no personal experience of graduate math departments) and because it seemed like a good answer to the question. If you (or anyone else) submits an answer, I'll still look at it, especially if you disagree with this one.2012-01-15
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    @CalvinMcPhail-Snyder : I gave an answer that (I think) is more reflective of mainstream thinking.2012-01-15
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    @james Firstly,it wasn't a "stab" at those students,it was an observation from my visits to NYU,Brown,Columbia and The Institute For Advanced Study at Princeton. Second-how is that a stab at the STUDENTS when the FACULTY and administrating officers of the university set the terms of the programs?!? Thirdly-more then one third of the people I've met at top graduate programs who aren't faculty were unfamiliar with generalized convergence and when I brought the matter up to several faculty there who were topologists,they dismissed it as "mere analysis and has nothing to do with modern topology."2012-01-15
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    @Adam You know,it's very sad that people reading our exchanges would take you as representative of the mathematical profession.That someone like you who doesn't know a damn thing about me or the people I know actually believes he's entitled to ridicule me in public.Do you realize that's the third straight time you've demeaned me at this message board? And why?Because I don't ascribe to your Almighty Doctrines? You're trying your best to goad me into blowing up here for your entertainment.I suggest you find some other way to entertain yourself.2012-01-15
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    @Mathemagician1234, I have no intention of fighting you on this issue. I will note the following: the students are at these universities because they have the necessary skills and have worked very hard to get there. It follows, perhaps, that the admins don't need to detail every little piece of information for students like this. Nonetheless, it is my experience that nets are common knowledge --- this, of course, may be wrong. Let's also note that a third of the people you've met may only be a small number.2012-01-15
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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.2012-01-15
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    Last, I would like to note something which I liked in your answer which was related to OP's question: if one wanted to show a student why nets can be neat, one needs only to show them Tychonoff's theorem's proof (assuming they've seen it without nets first). Bredon's book on topology does a section on nets rather elegantly and ends with this.2012-01-15
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    @Adam: Set-theoretic topologists would, I think, be surprised to hear that they aren’t ‘researchers in topology’.2012-01-17
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    @BrianM.Scott : I'm not really a topologist, but I've attended a large number of topology conferences, and I've never met a set-theoretic topologist in the wild. I'm sure they exist, but my perception is that the field has not been very active in a long time. Actually, I guess there is Justin Moore at Cornell, but my guess is that he would call himself a set theorist rather than a topologist. Anyway, I was trying to explain to mathemagician1234 why mainstream topologists are pretty dismissive of point-set topology and generalized convergence -- it just isn't relevant to most of the subject.2012-01-17
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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*.2012-01-18
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    @BrianM.Scott : So there is a community of such people. Interesting. Out of curiosity, can you give me (recent) examples of that kind of topology interacting with the traditional concerns of geometry/topology (eg manifolds, complex/real varieties, etc) or other mainstream fields of mathematics?2012-01-18
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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.2012-01-18
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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.