Timeline for On the cardinality of perfect spaces with the countable chain condition
Current License: CC BY-SA 3.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| S Apr 24, 2015 at 10:13 | history | suggested | Hachino |
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| Apr 24, 2015 at 9:40 | review | Suggested edits | |||
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| Nov 22, 2014 at 1:47 | vote | accept | Santi Spadaro | ||
| Jun 20, 2013 at 20:17 | comment | added | Santi Spadaro | Nice try, Dave. Unfortunately, the answer to my question is no. | |
| Jun 20, 2013 at 20:16 | answer | added | Santi Spadaro | timeline score: 6 | |
| Jun 11, 2013 at 2:00 | comment | added | David Milovich | Any T_3 counterexample must have large pi-character, just as your T_2 counterexample does. If X is T_3, has all points G_delta, and has more than c points, then X has more than c RO sets. By the corollary to 2.37 in Juhasz' book, if X is T_3, ccc, and has more than c RO sets, then X has an open set of points all with pi-character at least c^+. I speculate that if X is also T_{3.5}, then you might get some traction by working with a compactification of X, as most of the interesting consequences of large pi-character are for compact spaces. | |
| May 14, 2013 at 18:44 | comment | added | Mathieu Baillif | Thanks for this result, Santi. I was not aware of that. | |
| May 14, 2013 at 0:03 | comment | added | Santi Spadaro | Thank you, Mathieu, these are two very nice results. In fact, every first countable ccc space has cardinality at most continuum in ZFC, by an old result of Hajnal and Juhasz. | |
| May 13, 2013 at 23:23 | comment | added | Mathieu Baillif | Also, I just saw a paper by Larson and Tall ("Locally compact perfectly normal spaces may all be paracompact") in which their Theorem 2 states that is is consistent with ZFC that every first countable hereditarily normal countable chain condition space is hereditarily separable. | |
| May 13, 2013 at 22:53 | comment | added | Mathieu Baillif | I don't know if this helps, but a related result is that MA + nonCH implies that a locally compact first countable ccc space is separable, and thus has cardinality at most the continuum. I think it is due to I. Juhasz. Ref if needed: I. Juhasz: Cardinal functions in Topology. Number 34 in Mathematical Centre Tract. Mathematisch Centrum, 1971. | |
| May 13, 2013 at 21:48 | history | edited | Santi Spadaro | CC BY-SA 3.0 |
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| May 13, 2013 at 21:43 | history | asked | Santi Spadaro | CC BY-SA 3.0 |