Talk:Subnet (mathematics)

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[edit] Cofinal or cofinal increasing

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.

  • Wilard, Stephen (2004). General Topology. Dover Publications. ISBN 0486434796. 

--Kompik 10:46, 28 October 2007 (UTC)

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)

However, although my fear that we have a wrong definition, was unsubstantiated, it is still good to notice that we have found two definition in the literature. One of them defines a subnet via cofinal map (Engelking and Kelley don't use the name "cofinal" but their definition is in fact the same) and another approach requires cofinal and order-preserving (I have mentioned Willard, another book using the same definition for subnets is Beer: Topologies on Closed and Closed Convex Sets). Maybe this should be mentioned in the article?

In my opinion, although these two definitions are not equivalent, they are "compatible" in the sense that if we define a cluster point of net as a limit of some of its subnets, then we obtain the same notion of cluster point (taking either of these two definitions). --Kompik 22:06, 31 October 2007 (UTC)

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)
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)
I have gathered [1] 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)
Ok, so I have moved it here User:Kompik/Math/Subnet. --Kompik 11:43, 2 November 2007 (UTC)
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)
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)
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)
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)

[edit] Subnets of subsequences

The last sentance of the article is misleading in my opinion:

All subnets of a given sequence can be obtained by repeating its terms and reordering them.[2]

This is implying that there is a linear order to the elements of a subnet of a subsequence, which is false as it need only be indexed by an arbitrary directed set. Thus I am removing it. --Ray andrew (talk) 20:17, 2 May 2008 (UTC)

Thanks for noticing my mistake. I will correct the formulation. Here I include the full quotation from the book Werner Gähler: Grundstrukturen der Analysis I, Akademie - Verlag, Beriln, 1977 (in German).
Im weiteren geben wir eine Charakterisierung der Folgen an, die Tielnetze von Folgen sind. Wir benötingen zwein Definitionen: Sind x_1,x_2,\ldots und y_1,y_2,\ldots Folgen und existiert eine eineindeutige Abbildung f von \mathbb{N}^+ auf sich mit yn = xf(n) (n\in\mathbb{N}^+) so nennen wir y_1,y_2,\ldots eine Umordnung von x_1,x_2,\ldots, existiert eine (bezüglich der natürlichen Ordnung von \mathbb{N}^+) isotone Abbildung f von \mathbb{N}^+ auf sich mit yn = xf(n), so nennen wir y_1,y_2,\ldots eine Streckung von x_1,x_2,\ldots Umordnungen ergeben sich somit durch eventuelle Vertauschung der Reihenfolge der Glieder der betreffenden Folge und Streckungen, indem die Glider der Folge eventuell hintereinander mehrfach angeführt werden.

Satz 2.8.3 [Charakterisierung der Folgen, die Teilnetze von Folgen sind]

Eine Folge y_1,y_2,\ldots ist genau dann ein Teilnetz einer Folge x_1,x_2,\ldots, wenn sie eine Umordnung einer Streckung einer Teilfolge von x_1,x_2,\ldots ist.
My translation:
A characterization of sequences which are subnets of a given sequence will follow. We will use two definitions: If x_1,x_2,\ldots and y_1,y_2,\ldots are sequences and there exists a one-to-one map f from \mathbb{N}^+ to \mathbb{N}^+ with yn = xf(n) (n\in\mathbb{N}^+) we call y_1,y_2,\ldots a permutation of x_1,x_2,\ldots, if there exists a monotone map (w.r.t.~the usual ordering of \mathbb{N}^+) isotone f from \mathbb{N}^+ to \mathbb{N}^+ with yn = xf(n), then we call y_1,y_2,\ldots a dilation of x_1,x_2,\ldots Permutations are given by reordering the terms of a sequence and dilations by repeating them several times.

Theorem 2.8.3 [A characterization of sequences that are subnets of sequences]

A sequence y_1,y_2,\ldots is a subnet of set x_1,x_2,\ldots if and only if it is a permutation of dilatation of a subsequence of x_1,x_2,\ldots