Timeline for Are nets and filters useful in geometry and topology?
Current License: CC BY-SA 2.5
9 events
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| Jan 17, 2023 at 9:43 | comment | added | Jochen Glueck | @RachidAtmai: Hmm, I'm twelve and a half years late to the party - but as this just popped up at the front page again, I'd like to mention that proving Tychonoff's Theorem using nets is just as easy as the ultrafilter proof: one just needs the concept of universal nets, which is the "net analogue" of ultrafilters and can be used to characterize compactness in a similar way: a topological space is compact iff every universal net converges. As an analyst, I'm inclined to add that, once one has this, many compactness results in functional analysis become almost obvious. | |
| Jul 17, 2010 at 9:21 | comment | added | Yemon Choi | @alephomega: sometimes ultrafilters are cleaner; sometimes nets are more convenient or more intuitive (approximation arguments in biduals of Banach spaces or algebras, for instance). It depends on what you are trying to do. | |
| Jul 17, 2010 at 7:47 | comment | added | Rachid Atmai | I think nets are horrendous to use. Try proving Tychonoff Theorem using nets: it is just painful. When you use ultrafilters it is a much nicer proof. | |
| Jul 16, 2010 at 0:27 | comment | added | Yemon Choi | The notion of subnet being subtler than that of subsequence could be one obstacle, perhaps. On the other hand, the book of Beardon mentioned by Michael Greinecker (mathoverflow.net/questions/32035/… ) might represent a step in the direction you suggest. | |
| Jul 15, 2010 at 23:29 | comment | added | Charles Staats | The problem is that one of the axioms for a topology defined by nets, the diagonalization axiom or some such, is quite ugly and not too easy to work with (see Willard's book General Topology). However, I think it would be reasonable to define nets early on, and prove the sorts of theorems that are found in Munkres, Supplementary Exercises to Chapter 3 (leading up to "a space is compact iff every net has a convergent subnet). | |
| Jul 15, 2010 at 21:51 | comment | added | David Corwin | Interesting. Might anyone suggest that topology should be taught by introducing nets at the beginning? (i.e. reform the way that topology is taught) | |
| Jul 15, 2010 at 18:34 | comment | added | Matthew Daws | This sort of thing, in some sense, comes down to "taste": sure, any proof using nets can be reformulated using "bare hands" topology (i.e. open sets, inverse images etc.) but really this is just a formal process. I happen to think that nets make proofs a lot easier to follow, which leads to a better intuition, which in turn perhaps leads to new proofs which would have been impossible to see without the simplifications which nets provided. | |
| Jul 15, 2010 at 17:08 | comment | added | David Corwin | Interesting. Now is it just a useful tool to make a few lemmas easier to prove (like the ones outlined)? Are there deep results which use the net/ultrafilter formulation? | |
| Jul 15, 2010 at 17:06 | history | answered | Henno Brandsma | CC BY-SA 2.5 |