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Borel equivalence relation
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Borel equivalence relation on a <a href="/facts/Polish_space/CkUFcGap">Polish space</a> <i>X</i> is an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> on <i>X</i> that is a <a href="/facts/Borel_algebra/jvSsfpkP">Borel</a> subset of <i>X</i> × <i>X</i> (in the <a href="/facts/Product_topology/0hg02IaX">product topology</a>). </p><p>Given Borel equivalence relations <i>E</i> and <i>F</i> on Polish spaces <i>X</i> and <i>Y</i> respectively, one says that <i>E</i> is <i>Borel reducible</i> to <i>F</i>, in symbols <i>E</i> ≤B <i>F</i>, if and only if there is a <a href="/facts/Borel_function/NgIHnb1b">Borel function</a> </p> Θ : <i>X</i> → <i>Y</i> <p>such that for all <i>x</i>,<i>x</i>' ∈ <i>X</i>, one has </p> <i>x</i> <i>E</i> <i>x</i>' ⇔ Θ(<i>x</i>) <i>F</i> Θ(<i>x</i>'). <p>Conceptually, if <i>E</i> is Borel reducible to <i>F</i>, then <i>E</i> is "not more complicated" than <i>F</i>, and the quotient space <i>X</i>/<i>E</i> has a lesser or equal "Borel cardinality" than <i>Y</i>/<i>F</i>, where "Borel cardinality" is like <a href="/facts/Cardinality/m6VPojwT">cardinality</a> except for a definability restriction on the witnessing mapping. </p>