Talk:Filter (mathematics)

message: Concerning "filters on sets", I think T can be called "filter base"
without necessity of stability under intersection; it is only needed that T contains an element which is subset of the intersection of any two elements (and thus any finite intersection).
I think this issue "filters on sets" merits an extra page
(or a "filter base" page...), where more detailed discussion could take place.
MFH 00:02, 9 Mar 2005 (UTC)
replies: text: I agree, the article should be split. At the moment I lack the necessary knowledge to do the split myself so perhaps someone else should do it. MathMartin 19:48, 14 May 2005 (UTC)
thread_title: Filters on sets

message: # For every x, y in F, there is some element z in F, such that z ≤ x and z ≤ y. (F is a filter base)
* Wouldn't this definition also be the same if it was simplified to remove y and z ≤ y? If so, why not go with the simpler version?
No, it wouldn't. Otherwise it would be useless since for every x in F there would be x itself such that x ≤ x.  —Preceding unsigned comment added by 95.238.6.118 (talk) 23:50, 8 January 2011 (UTC)
thread_title:  need clarifications 

message: http://planetmath.org/encyclopedia/Filter.html defines:
A filter F is said to be fixed or principal if the intersection of all elements of F is nonempty; otherwise, F is said to be free or non-principal.
http://en.wikipedia.org/wiki/Filter_(mathematics)
The smallest filter that contains a given element p is a principal filter and p is a principal element in this situation. The principal filter for p is just given by the set {x in P | p <= x} and is denoted by prefixing p with an upward arrow.
Every filter principal in the sense of WikiPedia is principal in the sense of PlanetMath, but not vice verse.
We need to resolve this terminological issue. Porton 8:46, 3 Sep 2006
:The definition of "principal" given here on Wikipedia is the one I've always seen.  I'll look around and see if I can find out where this other one occurs. Michael Hardy 02:37, 4 September 2006 (UTC)
thread_title: PlanetMath defines principal filter differently

message: I thought I knew what a filter was; in fact, I was about to start merging the raft of PM articles into this one, so as to get a complete picture. But then, I lost my mind contemplating the following simple example from topology. I need help finding my mind.
Suppose the total space is X=R the real number line. Take as the filter base A=(0,1) the unit interval.  Now, according to the definition of a filter F, if A\subset B \subset X then B\in F. So the filter contains all the sets that contain A, right? 
Well, consider the set U= (-0.5, 1.5) \cup (2, 2+\epsilon). Now U is a perfectly valid subset of R, and A\subset U is true, right?  So, according to the definition, U must belong to the filter. See the problem? Its bad: U= (-0.5, 1.5) \cup (x-\epsilon, x+\epsilon) is an element of the filter, for any x and epsilon... Surely this is not the intent of the definition, but I don't see a way out.
One way out would be to insist that if A\subset B \subset X and if B is connected, then B\in F, but then one must define connectedness... 
Hopefully I'll snap out of my funk shortly, but at the moment, I am confused. Help appreciated.  linas 15:27, 27 November 2006 (UTC)
replies: text: But this is the intent of the definition. What makes you think otherwise? --Zundark 15:59, 27 November 2006 (UTC)
replies: text: Yow! Right. OK, I get it now. I've been visualizing this thing "upside-down" all this time! So it turns out that my whirlwind review of all things topological is actually a good thing (for me). Thanks. linas 19:28, 27 November 2006 (UTC)
thread_title:  Help! 

message: I think some of the text needs to be fixed because it's not readable on all computers.  Throughout there are less than or equal to signs that are typed directly, and another one that just appears as a block to me.  What should be used instead is the HTML symbol like &[number]; or LaTeX.  I'd make the changes myself, but I'm not 100% sure what symbol the squares are.
--132.170.156.99
:Using the &[number]; format wouldn't help - your browser must understand UTF-8 (otherwise it wouldn't show the less-than-or-equal-to signs correctly), so the problem must be that it can't find the symbol in any font, and encoding the symbol differently wouldn't change that. So you would have to use LaTeX, or rewrite things to avoid the use of the symbols. --Zundark 16:58, 11 June 2007 (UTC)
thread_title:  Math Formatting 

message: In the section on convergent filter bases, the variable A appears without any apparent introduction.  Dfeuer (talk) 01:55, 27 November 2007 (UTC)
thread_title:  Variable from nowhere 

message: I made two changes in Section 2 of this article.  First, I believe that the term limit point is dangerously overloaded, so I removed one of usages of limit point in this article: if a filter base F converges to x, the article now calls x a limit of F, not a limit point.  I hope others will agree that this is helpful for disambiguation.
Secondly, I expanded a statement recently added by User:OdedSchramm about characterizing closures in terms of filters and filter bases.  I also included the proofs of the equivalence, since they are easy and enlightening.  The formatting of this is not horrible but could be improved, I think. Plclark (talk) 03:33, 11 July 2008 (UTC)
thread_title:  Two changes 

message: Some authors argue that at least the notion of filterbase had been used by Vietoris before Cartan -- see here User:Kompik/Math/Filterhistory for some references. Would it be more correct to say that the notion of convergence along a filter was introduced by Cartan? --Kompik (talk) 12:25, 16 August 2009 (UTC)
thread_title:  Origin of the notion 

message: Reference to my book was removed as a "self-promoting reference".
I disagree that it should be removed.
My book is arguably the best (not a fact, but it's a fact that it is longest and most detailed) reference on the topic of this article.
If the book were written not by myself, I would anyway vote to have it in the list of references for this article. It is objectively a good reference on the topic.
Please discuss if it's possible to add the reference back.  — Preceding unsigned comment added by VictorPorton (talk • contribs) 21:44, 7 September 2017 (UTC)
replies: text: Wikipedia's policy on spam includes "adding references with the aim of promoting the author or the work being referenced," which this is probably a case of. The guidelines for scholarly sources also suggest it should not be included: it's not "vetted by the scholarly community" in the sense described. It is also a self-published source, which is strongly discouraged. Antonfire (talk) 13:43, 8 September 2017 (UTC)
replies: text: The book is self-published but the part of the article concerning filters is thoroughly based on a peer-reviewed article. --VictorPorton (talk) 14:39, 8 September 2017 (UTC)
replies: text: It does not appear from the edit history that the article is based on that reference. The reference was added by  in 2012, years after most of the article was written, without any other accompanying edits. It was removed in 2014. Besides this, a google search suggests that IJPAM may be a "predatory journal" in the sense described in the guidelines for scholarly sources, in which case that reference should also be treated as self-published. For example, IJPAM's publisher, Academic Publications, Ltd., appears in Beall's List of Predatory Journals and Publishers. - Antonfire (talk) 15:26, 8 September 2017 (UTC)

text: Looking through the edit history, another work by Victor Porton was added to the article (by VictorPorton) in 2012 and removed in 2014. Antonfire (talk) 14:07, 8 September 2017 (UTC)

text: The references section is not a list of good books on the topic, it is a list of reliable sources for the information in the article. To warrant inclusion, it would at the very least need to meet the guidelines for scholarship and for self-published sources. This source is unlikely to meet those guidelines no matter what you do; even if you establish yourself as an expert in the relevant field (which would have nothing to do with your follower counts on social media) there will be other sources for the same information which fit Wikipedia's standards better. It may warrant inclusion if it somehow becomes the standard reference among mathematicians on the topic (which I think is very unlikely), but then you will not need to worry about it because then someone else will include it anyway. You should take the time to carefully read the guidelines on self-promotion, particularly the point on "review your intentions". I think it is clear that you are here is mainly to get recognition for you and your book. - Antonfire (talk) 14:00, 9 September 2017 (UTC)
thread_title:  My book reference 

message: This is an excellently written article. I have read it carefully and can find no place where it would be improved by an inline citation.  — Preceding unsigned comment added by Chisq (talk • contribs) 18:00, 13 July 2018 (UTC)
thread_title:  Inline citations? 

message: the article states
:if N is a neighbourhood base at x and C is a filter base on X, then C → x if and only if C is finer than N.
This doesn't seem correct to me. A filter converges to x iff it (or even just a base) refines the full neighborhood system. But it is possible for a convergent base to not refine a judicious base for the neighborhood system. For example, let C be the eventuality filter for the sequence 1/n, for n ≥ 1 in the real numbers. Take for N a neighborhood base of 0, the intervals of the form (–1/n,1/n) for n ≥ 2. Then the set {1,1/2,1/3,...} is contained within no element of N, so C does not refine N. So we have one direction: a filter base C converges to x if it refines a neighborhood base. And we can say a filter base C converges to x if and only if it refines the full upward closed neighborhood system of x. But not if it just refines any small base. -lethe talk [ +] 04:15, 10 March 2020 (UTC)
thread_title:  filter converges iff it refines neighborhood base? 

message: In the motivation section we find the text
: 1. The empty set cannot contain anything so it will not belong to the filter.
But this doesn't seem to be implied by the definition given in the definition section. Instead we have
: A filter is proper if it is not equal to the whole set P. This condition is sometimes added to the definition of a filter.
So I guess the numbered point 1. above refers to proper filters, rather than to filters in general. If this guess is correct it should be easy to fix it. 
Alternatively, perhaps this point 1 is intended to say the filter itself should not be empty, rather than that it shouldn't contain the empty set? Then it would correspond to point 1 in the definition section.
Nathaniel Virgo (talk) 03:15, 14 July 2020 (UTC)
replies: text: . I agree, and changed property 1. in the motivating example accordingly; so now this example is parallel to the general definition. I considered adding a remark that the example will amount to a proper filter; however, on second thought, I believe including the empty set as the common root to start searching from may be useful in an implementation. - Jochen Burghardt (talk) 11:04, 14 July 2020 (UTC)
replies: text: My view is that we should use the definition by which filters are automatically "proper".  This is very standard, or at least its dual is very standard, when talking about ideals on the natural numbers.  Allowing a filter to be everything is one of those definitions that looks formally cleaner, but then forever after makes you add extra clauses, to no benefit.  It's similar to allowing 1 to be a prime number because it's divisible by only 1 and itself — that did use to be the way a fair number of mathematicians understood the primes, and it's superficially neater.  But the neatness isn't worth the trouble. --Trovatore (talk) 17:29, 14 July 2020 (UTC)
replies: text: I looked up the books I have available: Davey.Priestley.1990 (Def.9.1, p.184) use the distinction "filter / proper filter", like the article currently does. Birkhoff (p.25) defines an ideal dually to the article, and introduces in Exm.6 a "filter of sets" in the power set P(E) as "a dual ideal (not P(E) itself)"; this seems to agree with your suggestion for "filter of sets", but with the article for "ideal". Gratzer only defines ideals, using the distinction "ideal / proper ideal", dually to the article. However, all three books are in the context of lattice theory, rather than of arbitrary posets; this may have introduced some bias. - If you suggest to change current "proper filter" to new "filter", what would you change current "filter" to? - Jochen Burghardt (talk) 15:09, 15 July 2020 (UTC)
replies: text:  I wouldn't change current "filter" to anything at all, just as I don't have a name for \mathrm{PRIMES}\cup \{1\}.  Do we need to say anything about the current "filters", other than that some authors define filters that way and thereby introduce degenerate cases? --Trovatore (talk) 16:14, 15 July 2020 (UTC)
replies: text: Fun fact: In the recent past, 1 was considered a prime number by many (but not all) mathematicians. The quote that article: "Some other more technical properties of prime numbers also do not hold for the number 1 [...] By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit"." Mgkrupa 20:49, 9 July 2021 (UTC)

text: Suggestion: Rename this article to Filter (order theory) and create a new article called Filter (set theory) with most of this article's information about filters on sets moved there. Each article will be changed as needed so that the article "Filter (order theory)" can assume that a "filter" is allowed to be degenerate/non-proper (as is common in this field) while the article "Filter (set theory)" can assume that "filter" means proper filter (as is common in this field). Mgkrupa 20:41, 9 July 2021 (UTC)
thread_title:  Possible minor terminology conflict 

message: Per the discussions here and at Talk:Filters in topology, I copied content from those two articles (mostly from Filters in topology) to create the article Filter (set theory). That page had been a redirect to Filter (mathematics)#Filter on a set tagged with Template:R with possibilities.
I plan to move most of the information about filters on sets found in this article to the new article Filter (set theory) (some info will be kept) or to the article Filters in topology (for information related to topology). This article (Filter (mathematics)) will then be dedicated to the more general notion of (not-necessarily proper) filters on a poset. 
This will also solve this article's major problem of switching (often without mention) between requiring that "filters" be proper, and sometimes not requiring propriety (I will use dual ideal to refer to a filter that need not be proper). This article can now be dedicated to dual ideals while Filter (set theory) will be dedicated to proper filters on sets. 
Anyone have any comments? Complaints? Suggestions? Threats? Mgkrupa 20:21, 11 May 2022 (UTC)
replies: text: Thanks for looking at this, !  Unfortunately I don't think "dual ideal" solves the problem.  You have exactly the same problem in the dual case — some people allow X to be an element of the ideal, where X is the whole space, but (in my experience) most people don't, and it's less useful to allow it than not to allow it. --Trovatore (talk) 20:25, 11 May 2022 (UTC)
replies: text: I think that there's a misunderstanding. When I wrote: "(I will use dual ideal to refer to a filter that need not be proper)", I meant that I (personally) would use the term "dual ideal" in this section of this talk page - not in this article (i.e. Filter (mathematics)) itself. I was not proposing that references to non-proper filters in the article be replaced with the term "dual ideal". My apologies for not making that clear. In fact, I was proposing the opposite: "filter" should be a synonym of "dual ideal" throughout this whole article (referring to Filter (mathematics)). Mgkrupa 01:16, 14 May 2022 (UTC)
thread_title:  Proposal: Move this article to [[Filter (order theory)]] 

message: This article talks about filters that converge to a point. But presumably it is also just fine to talk about filters that converge to a closed set, right? One would still call these "convergent filters", right? They're just not built from neighborhoods of a point. 67.198.37.16 (talk) 01:32, 27 November 2023 (UTC)
thread_title:  Convergent filter