{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T07:17:35Z","timestamp":1649143055797},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6310,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,12]]},"abstract":"<jats:p>The study of infinitary Boolean operations was undertaken by the early researchers of descriptive set theory soon after Suslin's discovery of the important <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline1\" \/> operation. The first attempt to lay down their theory in a systematic fashion was the work of Kantorovich and Livenson [5], where they call these the <jats:italic>analytical operations<\/jats:italic>. Earlier, Hausdorff had introduced the <jats:italic>\u03b4<\/jats:italic><jats:italic>s<\/jats:italic> operations \u2014 essentially same as the <jats:italic>monotone<\/jats:italic><jats:italic>\u03c9<\/jats:italic>-ary Boolean operations, and Kolmogorov, independently of Hausdorff, had discovered the same objects, which were used in his study of the <jats:italic>R<\/jats:italic> operator.<\/jats:p><jats:p>The <jats:italic>\u03c9<\/jats:italic>-ary Boolean operations turned out to be closely related to most of the classical hierarchies over a fixed Polish space <jats:italic>X<\/jats:italic>, including, e. g., the Borel hierarchy (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline2\" \/>), the difference hierarchies of Hausdorff (<jats:italic>D<\/jats:italic><jats:sub>\u03b7<\/jats:sub>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline2\" \/>)), the <jats:italic>C<\/jats:italic>-hierarchy (<jats:italic>C<\/jats:italic><jats:sub>\u03be<\/jats:sub>) of Selivanovski, and the projective hierarchy (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline3\" \/>): for each of these hierarchies, every level can be expressed as the range of an <jats:italic>\u03c9<\/jats:italic>-ary Boolean operation applied to all possible sequences of open subsets of <jats:italic>X<\/jats:italic>. In the terminology of Dougherty [3], every level is \u201copen-<jats:italic>\u03c9<\/jats:italic>-Boolean\u201d (if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline4\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline5\" \/> are collections of subsets of <jats:italic>X<\/jats:italic> and <jats:italic>I<\/jats:italic> is any set, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline4\" \/> is said to be <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline5\" \/>-<jats:italic>I-Boolean<\/jats:italic> if there exists an <jats:italic>I<\/jats:italic>-ary Boolean operation <jats:italic>\u03a6<\/jats:italic> such that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline4\" \/> = <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline5\" \/><jats:sub>\u03a6<\/jats:sub>, i. e. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline4\" \/> is the range of <jats:italic>\u03a6<\/jats:italic> restricted to all possible <jats:italic>I<\/jats:italic>-sequences of sets from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001690X_inline5\" \/>). If in addition, the space <jats:italic>X<\/jats:italic> has a basis consisting of clopen sets, then the levels of the above hierarchies are also \u201cclopen-<jats:italic>\u03c9<\/jats:italic>-Boolean.\u201d<\/jats:p>","DOI":"10.2307\/2275817","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:59:54Z","timestamp":1146956394000},"page":"1287-1304","source":"Crossref","is-referenced-by-count":0,"title":["Boolean operations, Borel sets, and Hausdorff's question"],"prefix":"10.1017","volume":"61","author":[{"given":"Abhijit","family":"Dasgupta","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120001690X_ref003","first-page":"232","volume":"52","author":"Dougherty","year":"1987","journal-title":"Sequential discreteness and clopen-I-Boolean classes"},{"key":"S002248120001690X_ref002","unstructured":"Dasgupta A. , Studies in Borel sets, Ph.D. Thesis , University of California, Berkeley, 1994."},{"key":"S002248120001690X_ref001","first-page":"1","volume-title":"Proceedings of the symposia in pure mathematics","volume":"13","author":"Addison","year":"1974"},{"key":"S002248120001690X_ref004","volume-title":"Set theory","author":"Hausdorff","year":"1962"},{"key":"S002248120001690X_ref005","first-page":"214","article-title":"Memoir on the analytical operations and projective sets (I, II)","volume":"20","author":"Kantorovich","year":"1933","journal-title":"Fundamenta Mathematica"},{"key":"S002248120001690X_ref006","volume-title":"Topology","volume":"1","author":"Kuratowski","year":"1966"},{"key":"S002248120001690X_ref007","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120001690X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T19:59:17Z","timestamp":1557691157000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120001690X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,12]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1996,12]]}},"alternative-id":["S002248120001690X"],"URL":"https:\/\/doi.org\/10.2307\/2275817","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,12]]}}}