|
Borel Equivalence Relation
In mathematics, a Borel equivalence relation on a Polish space ''X'' is an equivalence relation on ''X'' that is a Borel algebra, 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 :''x'' ''E'' ''x''' ⇔ Θ(''x'') ''F'' Θ(''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. Kuratowski's theorem A measure space ''X'' is called a standard Borel space if it is Borel-isomorphic to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
