{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T11:47:36Z","timestamp":1774439256466,"version":"3.50.1"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":5980,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1997,9]]},"abstract":"<jats:p>We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.<\/jats:p><jats:p><jats:bold>\u00a71. Introduction<\/jats:bold>. For <jats:italic>X<\/jats:italic> a Polish (i.e., separable, completely metrizable) space and <jats:italic>E<\/jats:italic> a Borel equivalence relation on <jats:italic>X<\/jats:italic>, a (complete) classification of <jats:italic>X<\/jats:italic> up to <jats:italic>E<\/jats:italic>-equivalence consists of finding a set of invariants <jats:italic>I<\/jats:italic> and a map <jats:italic>c<\/jats:italic> : <jats:italic>X<\/jats:italic> \u2192 <jats:italic>I<\/jats:italic> such that <jats:italic>xEy<\/jats:italic> \u21d4 <jats:italic>c(x)<\/jats:italic> = <jats:italic>c(y)<\/jats:italic>. To be of any value we would expect <jats:italic>I<\/jats:italic> and <jats:italic>c<\/jats:italic> to be \u201cexplicit\u201d or \u201cdefinable\u201d. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.<\/jats:p><jats:p>The following partial (pre-)ordering is fundamental in organizing this study. Given equivalence relations <jats:italic>E<\/jats:italic> and <jats:italic>F<\/jats:italic> on <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic>, resp., we say that <jats:italic>E<\/jats:italic> can be <jats:italic>Borel reduced<\/jats:italic> to <jats:italic>F<\/jats:italic>, in symbols<\/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=\"S1079898600007514_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>if there is a Borel map <jats:italic>f<\/jats:italic> : <jats:italic>X<\/jats:italic> \u2192 <jats:italic>Y<\/jats:italic> with <jats:italic>xEy<\/jats:italic> \u21d4 <jats:italic>f(x)Ff(y)<\/jats:italic>. Then if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007514_inline1\"\/> is an embedding of <jats:italic>X\/E<\/jats:italic> into <jats:italic>Y\/F<\/jats:italic>, which is \u201cBorel\u201d (in the sense that it has a Borel lifting).<\/jats:p><jats:p>Intuitively, <jats:italic>E<\/jats:italic> \u2264<jats:sub><jats:italic>B<\/jats:italic><\/jats:sub><jats:italic>F<\/jats:italic> might be interpreted in any one of the following ways:<\/jats:p><jats:p>(i) The classi.cation problem for <jats:italic>E<\/jats:italic> is simpler than (or can be reduced to) that of <jats:italic>F<\/jats:italic>: any invariants for <jats:italic>F<\/jats:italic> work as well for <jats:italic>E<\/jats:italic> (after composing by an <jats:italic>f<\/jats:italic> as above).<\/jats:p><jats:p>(ii) One can classify <jats:italic>E<\/jats:italic> by using as invariants <jats:italic>F<\/jats:italic>-equivalence classes.<\/jats:p><jats:p>(iii) The quotient space <jats:italic>X\/E<\/jats:italic> has \u201cBorel cardinality\u201d less than or equal to that of <jats:italic>Y\/F<\/jats:italic>, in the sense that there is a \u201cBorel\u201d embedding of <jats:italic>X\/E<\/jats:italic> into <jats:italic>Y\/F<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/421148","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:10:37Z","timestamp":1146985837000},"page":"329-346","source":"Crossref","is-referenced-by-count":27,"title":["New Dichotomies for Borel Equivalence Relations"],"prefix":"10.1017","volume":"3","author":[{"given":"Greg","family":"Hjorth","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander S.","family":"Kechris","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600007514_ref006","unstructured":"[1995] Hjorth G. , Actions of S\u221e , circulated notes."},{"key":"S1079898600007514_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1994-1149121-0"},{"key":"S1079898600007514_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"S1079898600007514_ref020","unstructured":"[1996] Mazur K. , A modification of Louveau and Velickovic construction for F\u03c3 ideals , preprint."},{"key":"S1079898600007514_ref013","unstructured":"[1994] Kechris A. 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