{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T10:57:42Z","timestamp":1775645862951,"version":"3.50.1"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8228,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,9]]},"abstract":"<jats:title>Abtract<\/jats:title><jats:p>We prove that every countably determined set <jats:italic>C<\/jats:italic> is <jats:italic>U<\/jats:italic>-meager if and only if every internal subset <jats:italic>A<\/jats:italic> of <jats:italic>C<\/jats:italic> is <jats:italic>U<\/jats:italic>-meager, provided that the cofinality and coinitiality of the cut <jats:italic>U<\/jats:italic> are both uncountable. As a consequence we prove that for such cuts a countably determined set <jats:italic>C<\/jats:italic> which intersects every <jats:italic>U<\/jats:italic>-monad in at most countably many points is <jats:italic>U<\/jats:italic>-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We prove that the set <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200024087_inline1.png\"\/>, where <jats:italic>H<\/jats:italic> is an infinite integer, cannot be expressed as a countable union of countably determined sets each of which is <jats:italic>U<\/jats:italic>-meager for some cut <jats:italic>U<\/jats:italic> with min{cf(<jats:italic>U<\/jats:italic>), ci(<jats:italic>U<\/jats:italic>)} \u2265 \u03c9<jats:sub>1<\/jats:sub>. Also, every Borel, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200024087_inline2.png\"\/> or countably determined set <jats:italic>C<\/jats:italic> which is <jats:italic>U<\/jats:italic>-meager for every cut <jats:italic>U<\/jats:italic> is a countable union of Borel, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200024087_inline2.png\"\/> or countably determined sets respectively, which are <jats:italic>U<\/jats:italic>-nowhere dense for every cut <jats:italic>U<\/jats:italic>. Further, the class of Borel <jats:italic>U<\/jats:italic>-meager sets for min{cf(<jats:italic>U<\/jats:italic>),ci(<jats:italic>U<\/jats:italic>)} \u2265 \u03c9<jats:sub>1<\/jats:sub> coincides with the least family of sets containing internal <jats:italic>U<\/jats:italic>-meager sets and closed with respect to the operation of countable union and intersection. The same is true if the phrase \u201c<jats:italic>U<\/jats:italic>-meager sets\u201d is replaced by \u201c<jats:italic>U<\/jats:italic>-meager for every cut <jats:italic>U<\/jats:italic>.\u201d<\/jats:p>","DOI":"10.2307\/2275060","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:41:03Z","timestamp":1146955263000},"page":"906-914","source":"Crossref","is-referenced-by-count":3,"title":["<i>U<\/i>-meager sets when the cofinality and the coinitiality of <i>U<\/i> are uncountable"],"prefix":"10.1017","volume":"56","author":[{"given":"Bo\u0161ko","family":"\u017divaljevi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200024087_ref008","unstructured":"\u017divaljevi\u0107 B. , Every Borel function is monotone Borel (to appear)."},{"key":"S0022481200024087_ref007","volume-title":"Nonstandard analysis","author":"Robinson","year":"1966"},{"key":"S0022481200024087_ref006","first-page":"71","volume":"56","author":"Keisler","year":"1991","journal-title":"Meager sets on the hyperfinite time line"},{"key":"S0022481200024087_ref005","first-page":"1167","volume":"54","author":"Keisler","year":"1989","journal-title":"Descriptive set theory over a hyperfinite set"},{"key":"S0022481200024087_ref001","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1964-021-7"},{"key":"S0022481200024087_ref004","unstructured":"Jin R. , Cuts and U-Lusin sets in hyperintegers (to appear)."},{"key":"S0022481200024087_ref002","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1979-066-0"},{"key":"S0022481200024087_ref003","unstructured":"Henson C. W. and Ross D. , Analytic mappings on hyperfinite sets (to appear)."}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200024087","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T10:34:22Z","timestamp":1679481262000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200024087\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1991,9]]}},"alternative-id":["S0022481200024087"],"URL":"https:\/\/doi.org\/10.2307\/2275060","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,9]]}}}