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== comments ==
This used to be a redirect to [[analytical hierarchy]], but that doesn't make any sense as "analytical" is a lightface notion, whereas "projective" is boldface. This page and [[analytic set]] are candidates for a future merge into the [[pointclass]] page, when I get that written. --[[User:Trovatore|Trovatore]] 8 July 2005 06:19 (UTC)
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:Done. [[User:CMummert|CMummert]] · <small>[[User talk:CMummert|talk]]</small> 14:02, 31 March 2007 (UTC)
:I think it is nice to have a separation between X hierarchy and X set. For example:
:: [[Arithmetical hierarchy]] / [[Arithmetical set]]
:: [[Analytical hierarchy]] / [[Analytic set]]
:: [[Borel hierarchy]] / [[Borel set]] (= [[Borel algebra]])
:There is a little duplication of content, but I think it is helpful to a naive reader to start with the non-hierarchy definition and later learn about the stratification. [[User:CMummert|CMummert]] · <small>[[User talk:CMummert|talk]]</small> 14:06, 31 March 2007 (UTC)
::Well, you can make a case for that, but it does make maintenance and improvement more difficult. (By the way the "analytical hierarchy/analytic set" juxtaposition is wrong.) --[[User:Trovatore|Trovatore]] 16:37, 31 March 2007 (UTC)
:::The [[Lightface and darkface]] page is still unwritten. Not being a descriptive set theorist, I tend mentally identify the corresponding hierarchies.[[User:CMummert|CMummert]] · <small>[[User talk:CMummert|talk]]</small> 17:18, 31 March 2007 (UTC)
::::There's a [[pointclass]] page that treats that material, with redirects from [[lightface]], [[lightface pointclass]], [[boldface pointclass]], and a link from [[boldface (disambiguation)]]. No one seems to have touched that page but me. I think it's a critical concept, given that it's the essential subject matter of descriptive set theory (one could almost say it should bear the same relation to the [[descriptive set theory]] article that [[set]] bears to [[set theory]]). I think I did a decent start-class job on the article, but I wonder whether people are actually using the material, given that no one has edited it. --[[User:Trovatore|Trovatore]] 07:11, 1 April 2007 (UTC)
:::And thanks for reverting me at [[Analytic set]], I remembed the distinction when I added it to [[analytical hierarchy]] but not this morning. I wasn't thinking. [[User:CMummert|CMummert]] · <small>[[User talk:CMummert|talk]]</small> 18:27, 31 March 2007 (UTC)
== Relationship to the [[Borel hierarchy]]? ==
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.
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.
Another example: the diagram in the German page says that <math>\mathbf{\Sigma}^1_n, \mathbf{\Pi}^1_n\subset\mathbf{\Delta}^1_{n+1}</math>, which is equivalent to <math>\mathbf{\Sigma}^1_n, \mathbf{\Pi}^1_n\subset\mathbf{\Sigma}^1_{n+1}</math>. This means that every <math>\mathbf{\Sigma}^1_n</math> set and every <math>\mathbf{\Pi}^1_n</math> is the projection of a <math>\mathbf{\Pi}^1_n</math> set; while the latter implication is trivially true, I have no idea about the former even with ''n'' = 1. The inclusion gives us 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 in every uncountable Polish space? (We know that this is true for the first inclusion as stated [https://webusers.imj-prg.fr/~dominique.lecomte/Chapitres/6-Analytic%20and%20co-analytic%20sets.pdf here], of course [https://math.stackexchange.com/q/3638461 some choice may be needed]; [https://mathoverflow.net/q/425003 this post] may have addressed the second, although I don't know if it would work for every uncountable Polish space.)
What's more, the inclusion <math>\mathbf{\Pi}^1_n\subset\mathbf{\Delta}^1_{n+1}</math> tells us that <math>\mathbf{\Sigma}^1_n</math> sets are precisely the projections of <math>\mathbf{\Delta}^1_n</math> sets, so <math>\mathbf{\Delta}^1_n</math> sets are precisely those sets such that themselves as well as their projections are all <math>\mathbf{\Delta}^1_n</math> sets: by definition <math>\mathbf{\Delta}^1_n\subset\mathbf{\Sigma}^1_n</math>, and the projections of <math>\mathbf{\Sigma}^1_n</math> sets are also in <math>\mathbf{\Sigma}^1_n</math>. Conversely, a <math>\mathbf{\Sigma}^1_n</math> set is projection of <math>\mathbf{\Pi}^1_{n-1}</math> set, and the latter is a <math>\mathbf{\Delta}^1_n</math> set. (This works for ''n'' ≥ 2; for ''n'' = 1, the proposition "every analytic set is the projection of a <math>\mathbf{\Delta}^1_1</math> set" is something outside the hierarchy: we have to show that every analytic set is the projection of a Borel set, and <math>\mathbf{\Delta}^1_1</math> sets are precisely Borel sets ([[Lusin's separation theorem|Suslin's theorem]]).) [[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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