![](/files/happy5.png)
Co-separability of filter objects (Solved)
![$ a $](/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png)
![$ b $](/files/tex/b94226d9717591da8122ae1467eda72a0f35d810.png)
![$ U $](/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png)
![$ a\cap b = \{U\} $](/files/tex/d629028c6ede583e9f727914a84518040302cf11.png)
![$$\exists A\in a,B\in b: (\forall X\in a: A\subseteq X \wedge \forall Y\in b: B\subseteq Y \wedge A \cup B = U).$$](/files/tex/383c8f7cfee251bbc6606fbb6a41fbb185d817dc.png)
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.
Bibliography
*Victor Porton. Open problem: co-separability of filter objects
* indicates original appearance(s) of problem.
Your example is wrong
The set of all infinite sets of integers is not a filter. For example .
I haven't read your comment further.
Correction
Sorry, I was too hasty. What I meant is that is a "nontrivial ultrafilter" (wikipedia page ultrafilter) calls this "non-principal ultrafilter".
No counterexamples, it is proved
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?
--
Victor Porton - http://www.mathematics21.org
Is it really?
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.
Oh, my mistake
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).
A counterexample?
From the link it seems you have proved the result. What about the following what seems to be a counterexample?
Now there is no set
that would be minimal in
...