I wonder that how Suslin got the idea "analytic set" and the advanced knowledge in this field.
Can anyone explain this simply or recommend some reference?
All discussions and comments are welcome.
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4$\begingroup$ As currently phrased, the answer is trivial[1]. Maybe you mean something else? [1]: Just take $\{(p,0^\omega) \mid p \text{ codes an ill-founded tree}\}$. $\endgroup$– ArnoCommented Sep 28, 2020 at 12:57
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4$\begingroup$ 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. $\endgroup$– Alexandre EremenkoCommented Sep 28, 2020 at 13:28
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9$\begingroup$ I think you're looking for a Borel set whose projection is not Borel. $\endgroup$– A. BailleulCommented Sep 28, 2020 at 14:00
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1$\begingroup$ 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 $\endgroup$– Alessandro CodenottiCommented Sep 28, 2020 at 17:01
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6$\begingroup$ 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 $\endgroup$– Alessandro CodenottiCommented Sep 28, 2020 at 17:04
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1 Answer
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I like the presentation of descriptive set theory found in Chapter 8 of
Cohn, Donald L., Measure theory., Boston, MA: Birkhäuser. ix, 373 p. (1993). ZBL0860.28001.
answered Sep 30, 2020 at 11:40