Open Problem Garden - Co-separability of filter objects - Comments

Oh, my mistake (re: Co-separability of filter objects)

porton — Fri, 10 Sep 2010 21:23:10 +0200

I made a mistake in the statement of the conjecture as published on OPG. I corrected the problem statement both on OPG and on my blog. It should be rather than .

Indeed the equivalent reformulations of the theorem are correct and my proof (of a more general statement than this theorem) is not affected by the above mentioned error.

Robert, you do not understand me because I introduced new notations (that up, down, Cor, etc.) You may wish to read my preprint about these things (filters on posets and generalizations).

Is it really? (re: Co-separability of filter objects)

Robert Samal — Fri, 10 Sep 2010 14:13:42 +0200

You require that , and my filter does not contain empty set. I'm trying to find a counter-example because either I misunderstand the statement of the theorem, or the theorem is false.

Some proofs just happen to have mistakes. Unfortunately, I don't understand yours, it apparently uses lot of notation (up, down, Cor, ...) that I'm unfamiliar with.

No counterexamples, it is proved (re: Co-separability of filter objects)

porton — Wed, 08 Sep 2010 23:14:54 +0200

Then take and (I do not require filters to be proper).

Robert, why you are trying to find a counter-example for a proved theorem?

Here is the proof.

--
Victor Porton - http://www.mathematics21.org

Correction (re: Co-separability of filter objects)

Robert Samal — Wed, 08 Sep 2010 17:18:21 +0200

Sorry, I was too hasty. What I meant is that is a "nontrivial ultrafilter" (wikipedia page ultrafilter) calls this "non-principal ultrafilter".

Your example is wrong (re: Co-separability of filter objects)

porton — Tue, 07 Sep 2010 23:29:32 +0200

The set of all infinite sets of integers is not a filter. For example .

I haven't read your comment further.

A counterexample? (re: Co-separability of filter objects)

Robert Samal — Tue, 07 Sep 2010 19:12:47 +0200

From the link it seems you have proved the result. What about the following what seems to be a counterexample?

(the set of integers), , the set of all infinite sets of integers

Now there is no set that would be minimal in ...

Co-separability of filter objects

porton — Sun, 29 Nov 2009 19:17:17 +0100

Author(s): Porton

Subject: Unsorted

Conjecture Let and are filters on a set and . Then

See here for some equivalent reformulations of this problem.

This problem (in fact, a little more general version of a problem equivalent to this problem) was solved by the problem author. See here for the solution.

Maybe this problem should be moved to "second-tier" because its solution is simple.