# Talk:Filter (mathematics)

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Field: Algebra

## 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.

## 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

1. 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)

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 befor 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 intoruced by Cartan? --Kompik (talk) 12:25, 16 August 2009 (UTC)