{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T01:08:54Z","timestamp":1775869734850,"version":"3.50.1"},"reference-count":16,"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":7132,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove that for every family<jats:italic>I<\/jats:italic>of closed subsets of a Polish space each<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019691_inline1\"\/>set can be covered by countably many members of<jats:italic>I<\/jats:italic>or else contains a nonempty<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019691_inline2\"\/>set which cannot be covered by countably many members of<jats:italic>I<\/jats:italic>. We prove an analogous result for<jats:italic>\u03ba<\/jats:italic>-Souslin sets and show that if<jats:italic>A<\/jats:italic><jats:sup>#<\/jats:sup>exists for any<jats:italic>A<\/jats:italic>\u2282<jats:italic>\u03c9<jats:sup>\u03c9<\/jats:sup><\/jats:italic>, then the above result is true for<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019691_inline3\"\/>sets. A theorem of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz's theorem due to Kechris. Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding \u201cbig\u201d closed sets inside<jats:italic>of<\/jats:italic>\u201cbig\u201d<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019691_inline1\"\/>and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019691_inline3\"\/>are consequences of our results.<\/jats:p>","DOI":"10.2307\/2275926","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:52:43Z","timestamp":1146955963000},"page":"1022-1031","source":"Crossref","is-referenced-by-count":20,"title":["Covering analytic sets by families of closed set"],"prefix":"10.1017","volume":"59","author":[{"given":"S\u0142awomir","family":"Solecki","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019691_ref015","first-page":"58\u201373","volume-title":"On the cardinality of sets of reals, Foundations of mathematics","author":"Solovay","year":"1969"},{"key":"S0022481200019691_ref010","first-page":"285\u2013301","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"42","author":"Louveau","year":"1985"},{"key":"S0022481200019691_ref007","first-page":"263\u2013288","article-title":"The structure of \u03c3-ideals of compact sets","volume":"301","author":"Kechris","year":"1987","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200019691_ref004","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200019691_ref001","first-page":"217\u2013239","article-title":"Ensembles d\u2019unicit\u00e9 et d\u2019unicit\u00e9 au sens large","volume":"37","author":"Debs","year":"1987","journal-title":"Annales de l\u2019Institut Fourier (Grenoble)"},{"key":"S0022481200019691_ref009","unstructured":"Louveau A. , circulated manuscript."},{"key":"S0022481200019691_ref002","first-page":"68\u201372","volume-title":"Two Remarks About Analytic Sets","volume":"1401","author":"Van Engelen","year":"1989"},{"key":"S0022481200019691_ref014","doi-asserted-by":"crossref","first-page":"330\u2013338","DOI":"10.2307\/44152277","article-title":"On Borel sets with small covers","volume":"18","author":"Petruska","year":"1992","journal-title":"Real Analysis Exchange"},{"key":"S0022481200019691_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-9964-7"},{"key":"S0022481200019691_ref012","doi-asserted-by":"publisher","DOI":"10.4064\/fm-55-2-139-147"},{"key":"S0022481200019691_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1993-1113695-5"},{"key":"S0022481200019691_ref011","first-page":"71\u2013125","article-title":"A proof of projective determinacy","volume":"2","author":"Martin","year":"1989","journal-title":"Journal of the American Mathematical Society, Journal of the American Mathematical Society"},{"key":"S0022481200019691_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/1998129"},{"key":"S0022481200019691_ref016","first-page":"85\u201387","article-title":"Approximation des sous-ensembles analytiques par I\u2019interieur","volume":"281","author":"Saint Raymond","year":"1975","journal-title":"Comptes Rendus des S\u00e9ances de l\u2019Acad\u00e9mie des Sciences, S\u00e9rie I. Math\u00e9matique"},{"key":"S0022481200019691_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1977-0450070-1"},{"key":"S0022481200019691_ref006","first-page":"413\u2013441","volume":"57","author":"Kechris","year":"1992","journal-title":"Descriptive set theory and harmonic analysis"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200019691","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T14:22:18Z","timestamp":1586874138000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200019691\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,9]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1994,9]]}},"alternative-id":["S0022481200019691"],"URL":"https:\/\/doi.org\/10.2307\/2275926","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,9]]}}}