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Borel equivalence relation
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<h2 id="kuratowskis-theorem">Kuratowski's theorem</h2> <p>A <a href="/facts/Measure_space/w9Dx4KY1">measure space</a> <i>X</i> is called a <a href="/facts/Standard_Borel_space/RRWXf2ES">standard Borel space</a> if it is Borel-isomorphic to a Borel subset of a Polish space. Kuratowski's theorem then states that two standard Borel spaces <i>X</i> and <i>Y</i> are Borel-isomorphic <a href="/facts/Iff/bYSxGJ66">iff</a> |<i>X</i>| = |<i>Y</i>|. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Hyperfinite_equivalence_relation/SCjLdVpX">Hyperfinite equivalence relation</a></li> <li><a href="/facts/Wadge_hierarchy/N0nqDjbg">Wadge hierarchy</a> – levels of complexity for sets of realsPages displaying wikidata descriptions as a fallback</li> <li><a href="/facts/Entourage_(topology)/GKumHlHo">Entourage (topology)</a> – Topological space with a notion of uniform propertiesPages displaying short descriptions of redirect targets</li></ul> <ul><li>Harrington, L. A.; A. S. Kechris; A. Louveau (Oct 1990). <a href="https://doi.org/10.2307%2F1990906">"A Glimm–Effros Dichotomy for Borel equivalence relations"</a>. <i>Journal of the American Mathematical Society</i>. 3 (2): 903–928. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1990906">10.2307/1990906</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1990906">1990906</a>.</li> <li><a href="/facts/Alexander_S._Kechris/XUP7SToo">Kechris, Alexander S.</a> (1994). <a href="https://archive.org/details/classicaldescrip0000kech"><i>Classical Descriptive Set Theory</i></a>. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94374-9.</li> <li>Silver, Jack H. (1980). "Counting the number of equivalence classes of Borel and coanalytic equivalence relations". <i>Annals of Mathematical Logic</i>. 18 (1): 1–28. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0003-4843%2880%2990002-9">10.1016/0003-4843(80)90002-9</a>.</li> <li><a href="/facts/Vladimir_Kanovei/E7PfffoX">Kanovei, Vladimir</a>; Borel equivalence relations. Structure and classification. University Lecture Series, 44. American Mathematical Society, Providence, RI, 2008. x+240 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4453-3</li></ul>