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332:. 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
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eine σ-Algebra." (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
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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.) --
792:
953:
873:
261:
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.
396:
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
1226:{\displaystyle \mathbf {\Delta } _{1}^{1}\subset \sigma (\mathbf {\Sigma } _{1}^{1})\subset \mathbf {\Delta } _{2}^{1}\subset \cdots \subset \mathbf {P} }
344:). 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. --
1799:
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1233:. Perhaps in fact each inclusion is strict in eveyy uncountable Polish space? (We know that this is true for the first inclusion as stated
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1234:
97:
58:
166:, but that doesn't make any sense as "analytical" is a lightface notion, whereas "projective" is boldface. This page and
33:
1716:
set" is something outside the hierarchy: we have to show that every analytic set is the projection of a Borel set, and
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page is still unwritten. Not being a descriptive set theorist, I tend mentally identify the corresponding hierarchies.
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may have addressed the second, although I don't know if it would work for eveyy uncountable Polish space.)
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is closed under countable intersection seems to be nontrivial for me. A proof would be well appreciated.
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sind abgeschlossen bezüglich abzählbarer
Durchschnitte und abzählbarer Vereinigungen, insbesondere ist
83:
1023:{\displaystyle \mathbf {\Sigma } _{n}^{1},\mathbf {\Pi } _{n}^{1}\subset \mathbf {\Sigma } _{n+1}^{1}}
943:{\displaystyle \mathbf {\Sigma } _{n}^{1},\mathbf {\Pi } _{n}^{1}\subset \mathbf {\Delta } _{n+1}^{1}}
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There's no sense in having both articles, and the "hierarchy" title better reflects the content. --
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
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1132:
set; while the latter implication is trivially true, I have no idea about the former even with
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1410:
sets are precisely those sets such that themselves as well as their projections are all
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I think it is nice to have a separation between X hierarchy and X set. For example:
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464:, but it seems that perhaps this should be pointed out in the opening paragraphs.
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1491:{\displaystyle \mathbf {\Delta } _{n}^{1}\subset \mathbf {\Sigma } _{n}^{1}}
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1301:{\displaystyle \mathbf {\Pi } _{n}^{1}\subset \mathbf {\Delta } _{n+1}^{1}}
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For example, an important fact from the German version: "Alle
Klassen
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is closed under countable union" would be trivial, but to prove that
1242:
740:. Since the image of union is the union of images, the implication "
1682:= 1, the proposition "every analytic set is the projection of a
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and the analytic hierarchy does not show up till much later, as
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is closed under countable union and intersection from the page
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15:
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We seem to have almost the same content on the page for the
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Another example: the diagram in the German page says that
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are all closed under countable union and intersection, so
1779:, North Holland 1987, ISBN 0-444-70199-0, Corollary 1F.2
1136:= 1. The inclusion gives us the inclusion of σ-algebras
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More information need to be mentioned in the article
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page that treats that material, with redirects from
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359:, I remembed the distinction when I added it to
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826:{\displaystyle \mathbf {\Sigma } _{n+1}^{1}}
526:{\displaystyle \Sigma _{n}^{1},\Pi _{n}^{1}}
363:but not this morning. I wasn't thinking.
170:are candidates for a future merge into the
19:
1743:{\displaystyle \mathbf {\Delta } _{1}^{1}}
1709:{\displaystyle \mathbf {\Delta } _{1}^{1}}
1667:{\displaystyle \mathbf {\Delta } _{n}^{1}}
1593:{\displaystyle \mathbf {\Sigma } _{n}^{1}}
1559:{\displaystyle \mathbf {\Sigma } _{n}^{1}}
1525:{\displaystyle \mathbf {\Sigma } _{n}^{1}}
1437:{\displaystyle \mathbf {\Delta } _{n}^{1}}
1403:{\displaystyle \mathbf {\Delta } _{n}^{1}}
1369:{\displaystyle \mathbf {\Delta } _{n}^{1}}
1335:{\displaystyle \mathbf {\Sigma } _{n}^{1}}
1057:{\displaystyle \mathbf {\Sigma } _{n}^{1}}
860:{\displaystyle \mathbf {\Sigma } _{n}^{1}}
767:{\displaystyle \mathbf {\Sigma } _{n}^{1}}
729:{\displaystyle \mathbf {\Sigma } _{1}^{1}}
692:{\displaystyle \mathbf {\Delta } _{n}^{1}}
624:{\displaystyle \mathbf {\Sigma } _{n}^{1}}
457:{\displaystyle \mathbf {\Sigma } _{1}^{1}}
423:{\displaystyle \mathbf {\Sigma } _{1}^{0}}
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1633:{\displaystyle \mathbf {\Pi } _{n-1}^{1}}
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1125:{\displaystyle \mathbf {\Pi } _{n}^{1}}
1091:{\displaystyle \mathbf {\Pi } _{n}^{1}}
774:is closed under countable intersection
658:{\displaystyle \mathbf {\Pi } _{n}^{1}}
49:
1342:sets are precisely the projections of
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95:This article is within the scope of
38:It is of interest to the following
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1800:Mid-priority mathematics articles
174:page, when I get that written. --
115:Knowledge:WikiProject Mathematics
1795:Start-Class mathematics articles
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1750:sets are precisely Borel sets (
590:{\displaystyle \Delta _{n}^{1}}
558:{\displaystyle \Delta _{n}^{1}}
355:And thanks for reverting me at
135:This article has been rated as
1186:
1166:
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162:This used to be a redirect to
1:
1764:06:07, 12 February 2024 (UTC)
474:20:28, 27 November 2023 (UTC)
109:and see a list of open tasks.
787:{\displaystyle \Rightarrow }
1248:What's more, the inclusion
1816:
374:18:27, 31 March 2007 (UTC)
308:17:18, 31 March 2007 (UTC)
281:16:37, 31 March 2007 (UTC)
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1640:set, and the latter is a
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1239:some choice may be needed
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349:07:11, 1 April 2007 (UTC)
330:boldface (disambiguation)
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1030:. This means that every
178:8 July 2005 06:19 (UTC)
141:project's priority scale
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98:WikiProject Mathematics
1777:Descriptive Set Theory
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295:Lightface and darkface
226:Arithmetical hierarchy
28:This article is rated
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1674:set. (This works for
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235:Analytical hierarchy
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164:analytical hierarchy
121:mathematics articles
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1775:Y.N. Moschovakis:
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34:content assessment
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1752:Suslin's theorem
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112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1789:Categories
342:set theory
314:pointclass
312:There's a
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1678:≥ 2; for
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1243:this post
346:Trovatore
340:bears to
318:lightface
278:Trovatore
248:Borel set
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176:Trovatore
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299:CMummert
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158:comments
139:on the
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