Timeline for A unique ultrafilter extending a union of filters?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 14, 2011 at 2:34 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Added hypothesis to lemma; edited body
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| Aug 14, 2011 at 1:18 | comment | added | Joel David Hamkins | Yes, I agree with that. | |
| Aug 14, 2011 at 1:07 | comment | added | Adam Bjorndahl | Sorry, that last statement is not right. What I meant was -- if $F \cup G$ is to have any chance of extending to a unique ultrafilter, then we'd have to insist that the ideals I and J be orthogonal, in which case, if they are also principle, then $F \cup G$ generates the improper filter. | |
| Aug 14, 2011 at 0:56 | comment | added | Adam Bjorndahl | Thank you for your insights. I'm still mulling over the details of your response, but I have a few comments right away. Your 1st observation can also be viewed topologically: in \beta\omega, regular closed sets can intersect in a singleton, provided that singleton is an isolated point. The second observation is new to me, and I'm very happy to have a less trivial answer to question 0. Finally, unless I'm mistaken, the first Theorem you state can be strengthened to state that <F \cup G> is in fact the improper filter. This also follows, I think, from the fact that complete BAs don't have gaps. | |
| Aug 14, 2011 at 0:34 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |