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We seem to have almost the same content on the page for the [[Borel hierarchy]] and if one blurs one's eyes, they cannot be told apart. Perhaps some clarifying distinction should be drawn. Well, I mean, the Borel hierarchy starts with <math>\mathbf{\Sigma}^0_1</math> and the analytic hierarchy does not show up till much later, as <math>\mathbf{\Sigma}^1_1</math>, but it seems that perhaps this should be pointed out in the opening paragraphs. [[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 20:28, 27 November 2023 (UTC)
== More information need to be mentioned in the article ==
For example, an important fact from the German version: "Alle Klassen <math>\Sigma^1_n, \Pi^1_n</math> und <math>\Delta^1_n</math> sind abgeschlossen bezüglich abzählbarer Durchschnitte und abzählbarer Vereinigungen, insbesondere ist <math>\Delta^1_n</math> eine σ-Algebra.<ref>Y.N. Moschovakis: ''Descriptive Set Theory'', North Holland 1987, ISBN 0-444-70199-0, Corollary 1F.2</ref>" (The German version of the pages uses the lightface symbols for both Borel hierarchy and projective hierarchy. ) If I understand it correctly, it says that families <math>\mathbf{\Sigma}^1_n</math> and <math>\mathbf{\Pi}^1_n</math> are all closed under countable union and intersection, so <math>\mathbf{\Delta}^1_n</math> is a σ-algebra. So we have the inclusion of σ-algebras <math>\mathbf{\Delta}^1_1\subset\sigma(\mathbf{\Sigma}^1_1)\subset\mathbf{\Delta}^1_2\subset\cdots\subset\mathbf{P}</math>. Perhaps in fact each inclusion is strict for any uncountable Polish space?
We know that <math>\mathbf{\Sigma}^1_1</math> is closed under countable union and intersection from the page [[analytic set]]. Since the image of union is the union of images, the implication "<math>\mathbf{\Sigma}^1_n</math> is closed under countable intersection <math>\Rightarrow</math> <math>\mathbf{\Sigma}^1_{n+1}</math> is closed under countable union" would be trivial, but to prove that <math>\mathbf{\Sigma}^1_n</math> is closed under countable intersection seems to be nontrivial for me. A proof would be well appreciated. [[Special:Contributions/129.104.241.214|129.104.241.214]] ([[User talk:129.104.241.214|talk]]) 06:07, 12 February 2024 (UTC)
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