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Basis Problem for Turbulent Actions II: c 0 ‐Equalities
Author(s) -
Farah Ilijas
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012636
Subject(s) - mathematics , equivalence relation , borel equivalence relation , countable set , isomorphism (crystallography) , equivalence (formal languages) , combinatorics , polish space , bounded function , sequence (biology) , discrete mathematics , borel measure , mathematical analysis , probability measure , separable space , chemistry , biology , crystal structure , genetics , crystallography
Let ( X n , d n ) be a sequence of finite metric spaces of uniformly bounded diameter. An equivalence relation D on the product ∏ n X n defined byx →Dy →if and only if lim sup n d n ( x n , y n ) = 0 is a c 0 ‐equality . A systematic study is made of c 0 ‐equalities and Borel reductions between them. Necessary and sufficient conditions, expressed in terms of combinatorial properties of metrics d n , are obtained for a c 0 ‐equality to be effectively reducible to the isomorphism relation of countable structures. It is proved that every Borel equivalence relation E reducible to a c 0 ‐equality D either reduces a c 0 ‐equality D ' additively reducible to D , or is Borel‐reducible to the equality relation on countable sets of reals. An appropriately defined sequence of metrics provides a c 0 ‐equality which is a turbulent orbit equivalence relation with no minimum turbulent equivalence relation reducible to it. This answers a question of Hjorth. It is also shown that, whenever E is an F σ ‐equivalence relation and D is a c 0 ‐equality, every Borel equivalence relation reducible to both D and to E has to be essentially countable. 2000 Mathematics Subject Classification : 03E15.