{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T02:59:20Z","timestamp":1776999560768,"version":"3.51.4"},"reference-count":13,"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":3114,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2005,9]]},"abstract":"<jats:p>The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by \u221d(<jats:italic>f<\/jats:italic>) the separation rank of a real-valued function<jats:italic>f<\/jats:italic>in<jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>X<\/jats:italic>), with<jats:italic>X<\/jats:italic>a Polish space. Notice that in [3], Bourgain has provided a positive answer to this problem in the case of<jats:italic>K<\/jats:italic>satisfying<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200006745_inline1\"\/>with<jats:italic>X<\/jats:italic>a compact metric space. The key ingredient in Bourgain's approach is that whenever a sequence of continuous functions pointwise converges to a function<jats:italic>f<\/jats:italic>, then the possible discontinuities of the limit function reflect a local \u2113<jats:sup>1<\/jats:sup>-structure to the sequence (<jats:italic>f<jats:sub>n<\/jats:sub><\/jats:italic>)<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>. More precisely the complexity of this \u2113<jats:sup>1<\/jats:sup>-structure increases as the complexity of the discontinuities of<jats:italic>f<\/jats:italic>does. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A.S. Kechris and A. Louveau have invented the rank r<jats:sub><jats:italic>ND<\/jats:italic><\/jats:sub>(<jats:italic>f<\/jats:italic>) which permits the link between the<jats:italic>c<\/jats:italic><jats:sub>0<\/jats:sub>-structure of a sequence (<jats:italic>f<jats:sub>n<\/jats:sub><\/jats:italic>)<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>of uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal's<jats:italic>c<\/jats:italic><jats:sub>0<\/jats:sub>-theorem [11] and the<jats:italic>c<\/jats:italic><jats:sub>0<\/jats:sub>-index theorem [2] are consequences of this interaction.<\/jats:p><jats:p>Passing to the case where either (<jats:italic>f<jats:sub>n<\/jats:sub><\/jats:italic>)<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>are not continuous or<jats:italic>X<\/jats:italic>is a non-compact Polish space, this nice interaction is completely lost.<\/jats:p>","DOI":"10.2178\/jsl\/1122038909","type":"journal-article","created":{"date-parts":[[2005,7,22]],"date-time":"2005-07-22T18:43:10Z","timestamp":1122057790000},"page":"681-695","source":"Crossref","is-referenced-by-count":4,"title":["Tree structures associated to a family of functions"],"prefix":"10.1017","volume":"70","author":[{"given":"Spiros A.","family":"Argyros","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pandelis","family":"Dodos","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vassilis","family":"Kanellopoulos","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200006745_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/2373913"},{"key":"S0022481200006745_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0096295"},{"key":"S0022481200006745_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090209"},{"key":"S0022481200006745_ref003","first-page":"235","article-title":"On convergent sequences of continuous functions","volume":"32","author":"Bourgain","year":"1980","journal-title":"Bulletin de la Soci\u00e9t\u00e9 Math\u00e9matique de Belgique"},{"key":"S0022481200006745_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/S1874-5849(03)80030-X"},{"key":"S0022481200006745_ref008","first-page":"79","article-title":"Characterizations of spreading models of l1","volume":"41","author":"Kiriakouli","year":"2000","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"S0022481200006745_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"S0022481200006745_ref013","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-99-00312-4"},{"key":"S0022481200006745_ref011","first-page":"707","article-title":"A characterization of Banach spaces containing c0","volume":"7","author":"Rosenthal","year":"1994","journal-title":"Journal of the American Mathematical Society"},{"key":"S0022481200006745_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/s00208-002-0354-0"},{"key":"S0022481200006745_ref007","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1090\/S0002-9947-1990-0946424-3","article-title":"A classification of Baire class 1 functions","volume":"318","author":"Kechris","year":"1990","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200006745_ref009","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200006745_ref010","doi-asserted-by":"publisher","DOI":"10.2307\/2373824"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200006745","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T20:17:40Z","timestamp":1683145060000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200006745\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2005,9]]}},"alternative-id":["S0022481200006745"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1122038909","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,9]]}}}