Talk:Net (mathematics)

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Net stuff

Stock post message.svg To-do list for Net (mathematics): edit·history·watch·refresh· Updated 2013-01-15

  • Explain why a directed set is used and a partial order is insufficient (the article on Preorder claims that general directed sets rather than partially ordered ones are necessary, but that is not explained either here or there). Dfeuer 20:13, 12 Aug 2004 (UTC)
  • Figure out if I used the right terminology (should I have written cofinal rather than final?) Dfeuer 20:17, 12 Aug 2004 (UTC)
  • Describe the different definitions that authors have used for nets. Dfeuer 20:17, 12 Aug 2004 (UTC)
  • Prove that every net has a universal subnet. Dfeuer
  • Give an example of a discontinuous function which is sequentially continuous. Lupin 15:26, 6 Sep 2004 (UTC)

Who is H. L. Smith?[edit]

The original definition of nets appears in an article written by E. H. Moore and H. L. Smith. Who is the latter? My guess is Herman Lyle Smith, a PhD student of Moore according to the Mathematical Genealogy Project. Can anyone confirm this? -- Jitse Niesen 21:30, 26 Apr 2004 (UTC) Yes, Smith was Moore's student at University of Chicago. He went on to get a job at Louisiana State University and then dropped into obscurity. (See Halmos, Has progress in mathematics slowed down? American Mathematical Monthly, 97 (7), 1990.)

I added an article for Smith. It contains little more than the information above; Sundström says "Little biographical information about Smith is available. " She also cites the Halmos article. —Mark Dominus (talk) 18:59, 18 November 2010 (UTC)

Nets vs. Filters[edit]

According to Kelley, the equivalence of nets and filters is part of the folklore of the subject. Is there a formal equivalence? If not, the statement here should be modified.


Question[edit]

Under examples, you wrote Then Xs is a net. Would not Then (Xs) is a net be a little clearer given the previous section?

ε-nets[edit]

This article is about nets in topological spaces and not about ε-nets in metric spaces

The link ε-net has nothing to do with ε-nets in metric spaces, but I wasn't able to find the correct one. --Kompik 12:13, 5 November 2005 (UTC)

Nets vz. more general colimits[edit]

Directed preorders, or directed sets or directed filters, appear much more generally than just in topology, in some definitions of direct limits. I would like to know if the usage of the term net with meanings close to the one in the article also is more widespread. JoergenB 18:04, 15 November 2007 (UTC)

Intuitive interpretation[edit]

Instead of saying sequence do not encode enough "information" about continuity of function, I think it would be more appropriate to say sequences are not "long enough" to reach a certain point x. For example take the first ordinal w1 with the set of all smaller ordinals in order topology. w1 is a limit point of the set of smaller ordinals, but no sequence is "long" enough to "reach" w1, since every countable set has an upper bound strictly less than w1. However nets avoid this problem because they allow for much longer "sequences", might not be totally ordered ofc but still you get my ideaStandard Oil (talk) 13:54, 14 August 2009 (UTC)

You are indeed correct that your interpretation is very intutive in nature. Nevertheless, I do not know whether this would make sense to someone who does not know the concept well. In particular, I think that the distinction between nets and sequences is closely related to neighbourhood bases about points. Intuitively, if the basis about a point can be "linearly ordered in a countable manner", sequences will do. However, if not, one must analyse the behaviour of the neighbourhood base as a partially ordered set and determine an appropriate domain for a net. If you wish, you can create an intuitive section in the article (just below the lead), and write there, since your idea is not a bad one. --PST 01:51, 3 September 2009 (UTC)

Merger proposal[edit]

I am proposing the stub Cauchy net be merged into the section net (mathematics)#Cauchy nets. The reason is that both the article and the section are very short and their combination can help create a more complete treatment, and to centralize efforts to make them better and longer. Brent Perreault (talk) 04:53, 15 January 2013 (UTC)

Reducing information does not grant improvement[edit]

Due to recent shortenings the first paragraph is missing some information, e.g. there is some lonely "f" lying around whose definition only becomes clear by reading the diff to the previous version. Please, correct this part. 130.75.242.23 (talk) 17:24, 18 February 2013 (UTC)

The definition[edit]

The definition currently given on this page seems to disagree with the definition given by Kelley (pg. 70 of his General Topology), and I myself noticed difficulty in proving some things with the definition given here (and no problem with Kelley's definition).

Are we sure the definition given here is correct? — Preceding unsigned comment added by GleasSpty (talkcontribs) 00:20, 4 July 2015 (UTC)

Do you mean definition of net or definition of subnet? There might be several editions of Kelley's book, but in the one I have, net is defined on p. 65 and subnet is defined on p.70. Several definitions of subnet can be found in literature. See also the discussion at Talk:Subnet (mathematics). --Kompik (talk) 08:08, 4 July 2015 (UTC)
Yes. That was my mistake. I posted this question on the wrong talk page. The definition of net here is certainly correct. — Preceding unsigned comment added by GleasSpty (talkcontribs) 18:25, 5 July 2015 (UTC)

I think I found a mistake[edit]

when it is proved that a cluster point is the limit of a subnet, the pretendet sub net is not a subnet with repsect to the definition we find in wikipedia. In fact the map from B to A is not monotone.

This is a mistake.

Moreover, this mistake reflects in the subsequent theorem, when it is proved that a compact space is net-compact. If the proof whose correct, in the case A is countable one would get a proof that a compact space is sequentialli compact. Which is false. Indeed the infinite produt of [0,1] is compact by tychonoff but it is not sequentially compact.

Please, consider seriously mi concerns even if I wrote them in a very bad style.

If you think I'm wrong, please explain better why the proof proposed do not apply to the countable case.

Move proposal[edit]

I'm thinking we should move this to Net (topology) since there are other mathematical notions of nets, particularly Net (polyhedron), and I don't think either of them qualifies as primary. There are quite a few incoming links to change if this is done (which I'd be happy to take care of), but I wanted to make sure no one objected first. –Deacon Vorbis (carbon • videos) 18:33, 22 February 2018 (UTC)

See also ε-net for two more (mysteriously giving the appearance of three more, but really there's only two there). —David Eppstein (talk) 18:55, 22 February 2018 (UTC)
An alternative, completely unambiguous, title would be Moore–Smith convergence. Sławomir Biały (talk) 18:57, 22 February 2018 (UTC)
There is also a concept of net in combinatorics/finite geometry, so I agree that the current title is inappropriate. Either of the proposed changes is fine with me, but as net is one of those overused terms in mathematics I'd lean towards Moore−Smith.--Bill Cherowitzo (talk) 21:45, 22 February 2018 (UTC)

I think Net (mathematics) should be made a disambiguation page. Michael Hardy (talk) 00:39, 28 February 2018 (UTC)