Talk:Filter (mathematics)

The reference article about filters

I'm writing the text which is to become the exhaustive reference about filters on posets and filters on lattices.

The current PDF draft

Filters on sets

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)

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)

need clarifications

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)

The definition of ideal isn't really clear to me: "the concept obtained by reversing all ≤ and exchanging ∧ with ∨". In what context does it mean "all"? If it's just the general definition, the page on Ideal (order theory) doesn't look like it swapped the ∧ with ∨ in its definition of directed set.

TomJF 08:31, 19 April 2006 (UTC)

PlanetMath defines principal filter differently

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)

Help!

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)

But this is the intent of the definition. What makes you think otherwise? --Zundark 15:59, 27 November 2006 (UTC)

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)

Math Formatting

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)

Some Change Made

I feel that this page needs some work, and I began by making a few small changes. Ideally I would like to make some quite substantial changes -- most of all the discussion of filters on a set as a "special case" of filters on a poset seems unhelpfully general to me: there should be a page on filters on a set and another (presumably shorter) page describing the generalization to posets. The material on filters on a topological space should be carefully linked with the corresponding material on nets.

In the meantime there are several claims made here that I'm not sure I agree with and am temped to delete, but perhaps it is better to ask for justification. First, it seems wrong to say that filters are a generalization of nets -- in one sense they are equivalent to nets, and in another sense nets are the more general object: if one passes from a filter to a net and then back to a filter, one gets the original filter back again, but this is not the case for nets: the nets on a set form a proper class. The claim that a filter somehow comprises multiple nets seems similarly suspicious.

Also it is claimed that filters can be used to avoid the axiom of choice. I have never seen such a thing but could see how it might be true: can you provide a reference? -- P.L. Clark

Variable from nowhere

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

Two changes

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)Plclark

Origin of the notion

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)

My book reference

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)

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)

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)

It does not appear from the edit history that the article is based on that reference. The reference was added by VictorPorton 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)

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)

@VictorPorton: You might consider uploading your book to Wikimedia Commons instead, where the rules are less strict. Guessing from your book's title, commons:Category:Algebraic topology might be appropriate. - Jochen Burghardt (talk) 14:32, 8 September 2017 (UTC)

Well, now I ask for help. What are possible ways to make my book "vetted" by the scholarly community? I tried to publish it officially but several publishers rejected it as already self-published. I do not regret that I self-published the book, because it seems a greater way to disseminate the knowledge than "official" publication even by a big reputable publisher. What needs to happen to make my book "accepted" by the community? My blog has 100 subscribers. Plus 60 followers of book's Facebook page plus some number of Google+ subscribers. Add Twitter. And I have many more viewers which a not subscribers. With this number of subscribers I could not call myself an unrecognized genius :-) What specifically can I do for my book to be accepted by community? VictorPorton (talk) 11:21, 9 September 2017 (UTC)

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)

Inline citations?

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)

filter converges iff it refines neighborhood base?

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)

"Proper" filters

I'm used to the convention that an ideal on a set, by definition, does not include the whole set. This saves a lot of time with trivialities and doesn't seem to lose any useful generality. The dual convention here would be that filters are, by definition, "proper", which seems to be one of the attested conventions in the literature.

Should we consider rewriting the article to use that convention? I think it would be easier to read, because we wouldn't have to keep repeating the "proper subset" stuff. --Trovatore (talk) 00:05, 15 March 2020 (UTC)

This sounds good, but these problems could arise: (1) some examples might not work any longer (e.g. a Fréchet filter would be a filter only on an infinite set), and (2) there might be some algebraic structure on the set of all filters on a given poset X that would lose some properties (e.g. the filter X as a neutral element of some operation on filters). I can't judge whether (1) would apply to many examples, or/and (2) would apply at all; maybe some experts can comment on this. In any case, after a renaming we should have a name (e.g. "weak filter") for what is currently called "filter"; e.g. a renaming "proper filter"-->"filter" and "filter"-->"weak filter" would hardly cause unsolvable problems. You'd need to provide sources, anyway; how do they handle these issues? - Jochen Burghardt (talk) 09:56, 15 March 2020 (UTC)

I suspect they just ignore "weak filters", which aren't really good for anything that "proper filters" aren't good for. This is a similar question to whether there's a Boolean algebra with just a single element. The definition is a little cleaner if you allow it, but everything else just gets slightly sillier and for no benefit. --Trovatore (talk) 19:20, 15 March 2020 (UTC)

FYI Dugundji^[1] uses the term "dual ideal"^[2] for what you termed "weak filter". Also, the term "weak filter" seems to have been defined in this article^[3] from the 1990s as a (non-equivalent) generalization of "filter"; this definition of "weak filter" seems to still be in use, having appeared in this 2021 article.^[4] So I think that the term "dual ideal" should be used. Mgkrupa 16:52, 11 May 2022 (UTC)

Hi, Trovatore. Some observations:

(1) Looking through the article, it does not seem like this will be a major issue.

(2) You are rightfully worried about introducing false statements into the article. But there are already several there. For example, any family

B

that contains the empty set - such as the power set - converges to every point of a space

X

(because every neighborhood contains the empty set) and just below the article says

"Every limit of a filter base is also a cluster point of the base."

where "cluster at

x

" is correctly defined as: "if only if each element of

B

has non-empty intersection with each neighbourhood of x" which is certainly not true for

B

as it contains the empty set. There are other similar problems in the topology-related sections that result from allowing degenerate filters. My point is, regardless of whether or not the definition of "filter" is changed to require non-degeneracy, this article will require many corrections.

Also, the great majority of this article is about filters on sets, which from my experience are typically required to be non-degenerate, so:

Proposal: Create a new article: "Filter (order theory)"

to separate the order-theoretic results, which often allow degenerate filters, from those that are purely about filters on sets (e.g. those for topology). Mgkrupa 21:50, 27 May 2021 (UTC)

Major problem with article:

Because I'd never heard that term before, I google searched "proper filter base", which a term defined in this article. I got a grand total of 31 results. (Similarly, "proper prefilter" returns less than 2000 results, most of them related to air filters. Searching for "nondegenerate prefilter" gave 3 results - none math related; and although the exact words "proper prefilter" and "nondegenerate prefilter" is not used in this article, this is evidence that prefilters are always assumed to be non-degenerate). This is not a definition that is actually used. So despite the what this article says, prefilters/filter bases are required to be non-degenerate, which makes the definitions of "prefilter"/"filter base" given in this article FALSE. This major problem raises a second issue:

2nd point: The great majority of this article is about filters on sets and for this particular case, the power set is typically not considered a filter (which is consistent with how Henri Cartan first defined them). We should Wikipedia:Make technical articles understandable and "as accessible as possible to readers not already familiar with the subject matter" who are presumed to be new to the subject. Ultrafilters, Prefilters, and Filter subbases are never allowed to contain the empty set. So filters on sets being allowed to contain the empty set causes problems like the following (to give just one instance): a sentence such as "every filter is generated by a prefilter" (or one similar to it), which is commonly encountered "in the wild", would be FALSE because the power set is never generated by a prefilter. This might cause confusion for readers who aren't relying solely on this article for information. Mgkrupa 22:34, 27 May 2021 (UTC)

FYI 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). I will update these 2 articles appropriately later. That page had been a redirect to Filter (mathematics)#Filter on a set tagged with Template:R with possibilities. Mgkrupa 19:21, 11 May 2022 (UTC)

[1]
Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
[2]
Dugundji 1966, p. 213.
[3]
Schlechta, Karl (1 August 1995). "Defaults as Generalized Quantifiers". Journal of Logic and Computation. 5 (4): 473–494. doi:10.1093/logcom/5.4.473.
[4]
Koutras, Costas D.; Moyzes, Christos; Nomikos, Christos; Tsaprounis, Konstantinos; Zikos, Yorgos (20 October 2021). "On Weak Filters and Ultrafilters: Set Theory From (and for) Knowledge Representation". Logic Journal of the IGPL. doi:10.1093/jigpal/jzab030.

Possible minor terminology conflict

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)

Done. 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)

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)

I looked up the books I have available: Davey.Priestley.1990^[1] (Def.9.1, p.184) use the distinction "filter / proper filter", like the article currently does. Birkhoff^[2] (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^[3] 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)

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)

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)

[1]
Brian A. Davey and Hilary Ann Priestley (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press. ISBN 0-521-36766-2. LCCN 89009753.
[2]
Garrett Birkhoff (1967). Lattice Theory. Colloquium Publications. Vol. 25 (3rd ed.). Providence: Am. Math. Soc.
[3]
George Grätzer (2003). General Lattice Theory. Basel: Birkhäuser.

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)

Proposal: Move this article to Filter (order theory)

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)

Thanks for looking at this, Mgkrupa! 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)

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)

sounds good to me! —Sonarpulse ‡ Talk 05:11, 9 December 2024 (UTC)

Is there a mathematical use of "filter" that isn't part of order theory? --Trovatore (talk) 05:15, 9 December 2024 (UTC)

Simple examples needed

How about some really simple examples of filters and related concepts? The outside references (Rasiowa–Sikorski lemma, club filters and generic filters) are relatively complex. Consider the set X = {p,1,2,3}. (Where p is "just the letter" p.) What are all the filters on X? If too many then what are some of the filters on X? What are (some of) the filter bases of the filters? etc. Are there filters on N = {0,1,2,...}? Like what?

This whole topic is terminology-heavy and notation-dense. A few simple examples would help some of us better grasp the concepts. JohnH~enwiki (talk) 21:14, 9 March 2023 (UTC)

I don't have time to edit the page right now, but I'll note that the image at the top of the page gives examples of filters on a finite set. All filters on finite sets are "boring" principal filters like those. An example of a non-principal filter is the cofinite filter on an infinite set.

Caveat: I'm really only talking about Filter (set theory), which are really filters on the powerset lattice of a set X (i.e., all subsets of X ordered by inclusion). I'm not familiar with filters on general posets and I don't know if there exists useful examples of filters on a poset without a minimum element. Bbbbbbbbba (talk) 20:09, 10 March 2023 (UTC)

Thanks for pointing out the image. I'll update the examples section to mention it. JohnH~enwiki (talk) 14:24, 11 March 2023 (UTC)

Just a remark for now: your {p,1,2,3} example would need a partial order in the first place. - Jochen Burghardt (talk) 23:32, 10 March 2023 (UTC)

Oops, you're right. It was obvious to me but I didn't say it. I meant

1<2<3

but

p

doesn't compare to anything. JohnH~enwiki (talk) 14:16, 11 March 2023 (UTC)

I agree with the original poster: the example section is not really meant to be read by someone who is trying to get some intuition. For example, for me, a neighbourhood filter is the most intuitive example but maybe the article is not interested in examples from topology... —- Taku (talk) 12:53, 11 March 2023 (UTC)

By the way, reading the article it is not clear whether an ultrafilter is a filter or not. Of course, we know that but that obvious point shouldn’t be made?? —- Taku (talk) 13:07, 11 March 2023 (UTC)