Timeline for How to give a Borel set whose projection is not Borel?
Current License: CC BY-SA 4.0
15 events
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| Sep 30, 2020 at 11:40 | answer | added | Gerald Edgar | timeline score: 1 | |
| Sep 30, 2020 at 9:20 | comment | added | Alessandro Codenotti | @YOTAL look at theorem 14.2 in Kechris Classical Descriptive Set Theory, in this theorem an analytic but not Borel subset of $\mathcal N^2$ is constructed, which answers your question as well since any analytic subset of $\mathcal N^2$ is the projection of a Borel subset of $\mathcal N^3$ by definition | |
| Sep 30, 2020 at 7:59 | history | edited | bof | CC BY-SA 4.0 |
corrected title according to OP comment
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| Sep 30, 2020 at 6:20 | comment | added | YOTAL | @AlessandroCodenotti With all my gratitude, can you explain it more, or offer my some references? | |
| Sep 30, 2020 at 6:18 | history | edited | YOTAL | CC BY-SA 4.0 |
edited tags; edited title
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| Sep 30, 2020 at 6:17 | comment | added | YOTAL | @AlexandreEremenko Right. My fault | |
| Sep 30, 2020 at 6:15 | comment | added | YOTAL | @Arno Yes. Thank you for your answer. I mean a Borel set whose projection is not Borel | |
| Sep 30, 2020 at 6:14 | comment | added | YOTAL | @A.Bailleul Yes. Sorry, it is my fault. I'll modify my question. | |
| Sep 28, 2020 at 17:04 | comment | added | Alessandro Codenotti | As for how the notion of analytic sets was discovered that happened because Lebesgue wrote a paper in which he "proved" that the projection of a Borel set is Borel and Suslin noticed this mistake | |
| Sep 28, 2020 at 17:01 | comment | added | Alessandro Codenotti | And the way a Borel set whose projection is not Borel is usually constructed is by constructing a $\mathbf{\Sigma}^1_1$-universal set, which then cannot be Borel, because self-dual pointclasses are easily shown not to have universal sets | |
| Sep 28, 2020 at 14:00 | comment | added | A. Bailleul | I think you're looking for a Borel set whose projection is not Borel. | |
| Sep 28, 2020 at 13:46 | review | Close votes | |||
| Oct 7, 2020 at 3:06 | |||||
| Sep 28, 2020 at 13:28 | comment | added | Alexandre Eremenko | The example you ask is trivial: take to horizontal planes, and consider the set which consists of arbitrary set (non-Borel) in one of them and entire second plane. The vertical projection of this set is a plane. The example which led to discovery of analytic set is different: it is a Borel set whose projection is not Borel. | |
| Sep 28, 2020 at 12:57 | comment | added | Arno | As currently phrased, the answer is trivial[1]. Maybe you mean something else? [1]: Just take $\{(p,0^\omega) \mid p \text{ codes an ill-founded tree}\}$. | |
| Sep 28, 2020 at 12:49 | history | asked | YOTAL | CC BY-SA 4.0 |