{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T13:00:37Z","timestamp":1775826037713,"version":"3.50.1"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"<jats:p>Let \u0393 be a class of subsets of Baire space (<jats:sup>\u03c9<\/jats:sup>\u03c9) closed under inverse images by continuous functions. We say such a \u0393 is <jats:italic>continuously closed<\/jats:italic>. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_inline1\"\/>, the class dual to \u0393, consist of the complements relative to <jats:sup>\u03c9<\/jats:sup>\u03c9 of members of \u0393. If \u0393 is not selfdual, i.e., <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_inline2\"\/>, then let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_inline3\"\/>. A continuously closed nonselfdual class \u0393 of subsets of <jats:sup>\u03c9<\/jats:sup>\u0393 is said to have the <jats:italic>first separation property<\/jats:italic> [2] if<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>The set <jats:italic>C<\/jats:italic> is said to <jats:italic>separate A<\/jats:italic> and <jats:italic>B<\/jats:italic>. The class \u0393 is said to have the <jats:italic>second separation property<\/jats:italic> [3] if<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_eqnU2\"\/><\/jats:disp-formula><\/jats:p><jats:p>We shall assume the axiom of determinateness and show that if \u0393 is a continuously closed class of subsets of <jats:sup>\u03c9<\/jats:sup>\u03c9 and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_inline2\"\/> then<\/jats:p><jats:p>(1) \u0393 has the first separation property iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_inline1\"\/> does not have the second separation property, and<\/jats:p><jats:p>(2) either \u0393 or <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_inline1\"\/> has the second separation property.<\/jats:p><jats:p>Of course, (1) and (2) taken together imply that \u0393 and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049902_inline1\"\/> cannot both have the first separation property.<\/jats:p>","DOI":"10.2307\/2271950","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:46:34Z","timestamp":1146951994000},"page":"77-81","source":"Crossref","is-referenced-by-count":13,"title":["Separation principles and the axiom of determinateness"],"prefix":"10.1017","volume":"43","author":[{"given":"Robert A.","family":"van Wesep","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049902_ref006","volume-title":"Subsystems of analysis, and descriptive set theory under the axiom of determinateness","author":"Van Wesep","year":"1977"},{"key":"S0022481200049902_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-10-1-1-95"},{"key":"S0022481200049902_ref007","first-page":"714","article-title":"Degrees of complexity of subsets of the Baire space","volume":"19","author":"Wadge","year":"1972","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200049902_ref003","volume-title":"Le\u00e7ons sur les ensembles analytiques et leurs applications","author":"Lusin","year":"1930"},{"key":"S0022481200049902_ref005","unstructured":"Steel J. , Subsystems of analysis and the axiom of determinateness. Thesis, University of California, Berkeley, 1976."},{"key":"S0022481200049902_ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-26-1-183-191"},{"key":"S0022481200049902_ref004","unstructured":"Martin D. A. , The Wadge ordering is a well-ordering, unpublished manuscript."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049902","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T20:19:21Z","timestamp":1558988361000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049902\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1978,3]]}},"alternative-id":["S0022481200049902"],"URL":"https:\/\/doi.org\/10.2307\/2271950","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,3]]}}}