Timeline for $T_2$ topologies that are "as disjoint as possible"
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 24, 2016 at 7:50 | vote | accept | Dominic van der Zypen | ||
| Mar 23, 2016 at 16:33 | answer | added | Lajos Soukup | timeline score: 9 | |
| Mar 22, 2016 at 19:45 | vote | accept | Dominic van der Zypen | ||
| Mar 24, 2016 at 7:50 | |||||
| Mar 22, 2016 at 18:19 | answer | added | Lajos Soukup | timeline score: 10 | |
| Mar 22, 2016 at 16:47 | answer | added | Will Brian | timeline score: 13 | |
| Mar 22, 2016 at 16:23 | comment | added | Todd Trimble | @WillBrian Thanks! That's interesting. It might be good for someone to lay it all out in an answer, since not everyone seems to have access to the paper cited by Lajos. | |
| Mar 22, 2016 at 16:13 | comment | added | Will Brian | @ToddTrimble: You are right -- somehow I failed to notice. At any rate, the paper mentioned by Lajos still answers the question: Proposition 3.2 says that we do not get any such topologies on a countable set, and Corollary 3.4 says that (consistently) we do not get any such topologies on a set of size $\aleph_1$. So the paper shows that the answer to Dominic's first question is yes for some sets and no for some others, and for some we don't seem to know. | |
| Mar 22, 2016 at 15:37 | comment | added | Todd Trimble | I don't see how Will's first comment answers the question. Wasn't the problem to show this for any infinite set, not just some infinite set? | |
| Mar 22, 2016 at 15:25 | comment | added | Will Brian | @LajosSoukup: I encourage you to post your comment as an answer. It seems that Corollary 3.8 in the paper you found answers Dominic's first question. After a little poking around, I think the second question might be open (and reasonably difficult). | |
| Mar 22, 2016 at 15:00 | comment | added | Dominic van der Zypen | Thanks - can somebody of \{Lajos, Will\} post their comments in an answer? Of course it would be great to have an answer to the particular case $X=\mathbb{R}$, but your hints were already really helpful | |
| Mar 22, 2016 at 14:05 | comment | added | Will Brian | The paper mentioned by Lajos answers your first question: the authors show that there are two topologies on a set of size $2^{2^{\aleph_0}}$ (both are realizations of the space $\omega^* = \beta \omega - \omega$) whose intersection is precisely the cofinite sets. They don't seem to answer your second question, but there are some relevant-looking results, so you'll want to take a look. | |
| Mar 22, 2016 at 13:54 | comment | added | Todd Trimble | This paper might also be useful: sciencedirect.com/science/article/pii/S0166864111003336 I haven't read far enough into it to see whether it definitively answers your question. | |
| Mar 22, 2016 at 13:18 | comment | added | Dominic van der Zypen | Thanks a lot @LajosSoukup . I don't have access to that paper, do you know whether there is something on $T_2$-topologies in particular? | |
| Mar 22, 2016 at 13:13 | comment | added | Lajos Soukup | If $\tau_1$ and $\tau_2$ meet your requirement, then they are called $T_1$-independent. The following paper may contain useful information: Shakhmatov, D.; Tkachenko, M.; Wilson, R. G. Transversal and T1-independent topologies. Houston J. Math. 30 (2004), no. 2, 421–433. | |
| Mar 22, 2016 at 11:05 | comment | added | Dominic van der Zypen | @SimonHenry No worries - I missed that point, too... | |
| Mar 22, 2016 at 10:55 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
included cofinite sets
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| Mar 22, 2016 at 10:54 | comment | added | Dominic van der Zypen | Oh right - thanks for this hint! Will edit the question accordingly. | |
| Mar 22, 2016 at 10:48 | comment | added | Will Brian | Every $T_1$ topology contains every cofinite set, so being "as disjoint as possible" means at least including all of these. | |
| Mar 22, 2016 at 10:24 | comment | added | Dominic van der Zypen | Cool - can you jot down an example as an answer? | |
| Mar 22, 2016 at 8:44 | history | asked | Dominic van der Zypen | CC BY-SA 3.0 |