Talk:Filter (mathematics)

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[edit] 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)

[edit] 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?
  • 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)

[edit] 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)

[edit] 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)
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