Talk:Subnet (mathematics)

message: It would be good to include som reference for the definition. I'm not sure it is correct in the present form. The current version of article states, that a subnet is given by a cofinal function, where
: h is a cofinal function if for every α in A, there is a β in B such that for all γ ≥ β, h(γ) ≥ α.
or (as stated in Cofinal (mathematics))
:A cofinal function is a function f: X → A with preordered codomain A such that its range f(X) is cofinal in the codomain.
Willard's General topology defines subnet via cofinal and increasing non-decreasing function. Engelking's General topology defines finer net to (x_\sigma)_{\sigma\in\Sigma} using a map fulfilling the condition
:For every \sigma_0\in \Sigma there exists  a \sigma'_0\in \Sigma' such that \varphi(\sigma')\geq\sigma_0 whenever \sigma'\geq\sigma'_0.
The last condition is weaker than cofinal+increasingnon-decreasing, but stronger than the definition in the article. Therefore I think it would be good to provide a reference for the definition mentioned in article.
*
*
--Kompik 10:46, 28 October 2007 (UTC)
replies: text: I apologize -- of course the definition in the article is correct. The notion of cofinal function as defined above (which is different from the notion defined in the (current version of) article Cofinal (mathematics)) is precisely the notion corresponding to the definition I quoted from Engelking and Kelley. I have included the book Volker: A Taste of Topology into references - in this book I have found definition of subnet using the notion of cofinal map and the same definition of cofinal map as that one which is in the article. --Kompik 21:54, 31 October 2007 (UTC)

text: Since both definitions seem to be in use we should probably mention both of them. I would vote in favor of using Willard's definition (which seems to be the more modern one) as the primary definition and listing the present one as an alternative. Given a function h : B → A between directed sets one should at least hope that it be monotone. It seems to me that one should use the simplest definition of a subnet adequate to needs at hand. Unfortunately using cofinal subsets doesn't work, but using monotone, cofinal functions does. Another advantage is that Willard's usage of the term cofinal function agrees with the one given in our cofinal article (a function with a cofinal image). Another book using Willard's definition is Munkres' Topology (2000). -- Fropuff 17:10, 1 November 2007 (UTC)
replies: text: Well both definitions have their advantages. To me (but it's difficult to prove my claim without extensive checking of various textbooks) the Engelking's topology definition seems to be more frequent. Anyway, if you decide to change it to cofinal and monotone, I thing it's better to keep the current definition of cofinal function. At least for that reason, that then the two definitions can be formulated very easily: one of them uses function which is cofinal and monotone, the other one omits the word monotone. (If we understand cofinal function as the function with cofinal range, this is not true anymore.) As you may have noticed I mentioned this problem in Talk:Cofinal (mathematics) as  well. --Kompik 20:00, 1 November 2007 (UTC)

text: I have gathered   comparison of definition of subnet from several topology books. Maybe this will help. I hope this will not be understood as advertising my webpage -- converting this to wiki format would take some time. --Kompik 21:17, 1 November 2007 (UTC)

text: Ok, so I have moved it here User:Kompik/Math/Subnet. --Kompik 11:43, 2 November 2007 (UTC)
replies: text: I've changed the definition according to your suggestion (i.e., now the notion of cofinal map is the same as here). To be consisting we should be careful when changing some of the two articles. It seems that we have been discussing the same thing here and in Talk:Cofinal (mathematics), so it would be good to continue there. --Kompik 20:59, 10 November 2007 (UTC)

text: Eric Schechter's Handbook of Analysis and its Foundations has a good discussion of the different definitions of subnet, and their relationship.  He also suggests terminology to distinguish (something like Kelley subnet and Willard subnet).  —Preceding unsigned comment added by Waltpohl (talk • contribs) 06:26, 10 December 2007 (UTC)
replies: text: Unfortunately this book is not accessible to me; feel free to add those definitions at User:Kompik/Math/Subnet if you find it a good idea. --Kompik (talk) 12:00, 29 December 2007 (UTC)

text: Another book that gives the definition of a subnet not requiring monotonicity is "Real Analysis" 2nd ed. by Folland. Does anyone know of any result that aplies for one definition but not the other? --Ray andrew (talk) 20:17, 2 May 2008 (UTC)
thread_title:  Cofinal or cofinal increasing