|
In mathematics, a Borel equivalence relation on a Polish space ''X'' is an equivalence relation on ''X'' that is a Borel subset of ''X'' × ''X'' (in the product topology). ==Formal definition== Given Borel equivalence relations ''E'' and ''F'' on Polish spaces ''X'' and ''Y'' respectively, one says that ''E'' is ''Borel reducible'' to ''F'', in symbols ''E'' ≤B ''F'', if and only if there is a Borel function :''Θ'' : ''X'' → ''Y'' such that for all ''x'',''x' ''∈ ''X'', one has :''xEx' '' ⇔ ''Θ''(''x'')''FΘ''(''x' ''). Conceptually, if ''E'' is Borel reducible to ''F'', then ''E'' is "not more complicated" than ''F'', and the quotient space ''X''/''E'' has a lesser or equal "Borel cardinality" than ''Y''/''F'', where "Borel cardinality" is like cardinality except for a definability restriction on the witnessing mapping. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Borel equivalence relation」の詳細全文を読む スポンサード リンク
|