Timeline for Is $\mathcal{P}(\omega)/fin$ with the interval topology a connected space?
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 24, 2015 at 11:28 | comment | added | Ramiro de la Vega | @JoelDavidHamkins, of course I meant finitely many, no just two. | |
| Mar 24, 2015 at 5:50 | vote | accept | Dominic van der Zypen | ||
| Mar 24, 2015 at 1:01 | comment | added | Joel David Hamkins | I see. Meanwhile, I had some fun figuring it out, and I think it is a good problem. Regarding your remark, it doesn't seem to suffice in general for hyperconnectivity to show merely that nonempty sub-basic open sets intersect. For example, consider any Hausdorff space with three points fixed, and consider the sub-basis consisting of all open sets containing two out of three of those points. This generates the same topology, but notice that any two of these subbasic open sets intersect, even when there can be disjoint open sets. | |
| Mar 24, 2015 at 0:35 | comment | added | Ramiro de la Vega | @JoelDavidHamkins, I'm saying that there are no proper clopen subsets just because there are no disjoint non-empty open subsets (and this is a trivial reason). The fact that non-empty subbasic open sets intersect is not trivial but it is something that anyone interested in this space should be able to prove by himself. | |
| Mar 23, 2015 at 23:59 | comment | added | Joel David Hamkins | @RamirodelaVega, are you saying that it is trivial to see that any two nonempty open sets intersect? This is true, as I argue in my answer, but I found it to be a subtle issue. | |
| Mar 23, 2015 at 21:31 | comment | added | Ramiro de la Vega | There aren't, so the space is connected for trivial reasons. | |
| Mar 23, 2015 at 18:40 | answer | added | Joel David Hamkins | timeline score: 5 | |
| Mar 23, 2015 at 18:32 | comment | added | Dominic van der Zypen | I can't... But this doesn't mean there aren't such open sets. My intuition for the interval topology on this particular poset is quite poor. | |
| Mar 23, 2015 at 17:51 | comment | added | Ramiro de la Vega | Can you show us two disjoint non-empty open subsets? | |
| Mar 23, 2015 at 13:15 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
catchier formulation of title
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| Mar 23, 2015 at 10:21 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
added 77 characters in body
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| Mar 23, 2015 at 10:00 | history | asked | Dominic van der Zypen | CC BY-SA 3.0 |