A trichotomy for a class of equivalence relations  

作　　者：DING LongYun 

机构地区：[1]School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

出　　处：《Science China Mathematics》2012年第12期2621-2630,共10页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.11071129);the Program for New Century Excellent Talents in University (Grant No.09-0477)

摘　　要：Let Xn, n ∈ N be a sequence of non-empty sets, ψn : Xn2 → IR+. We consider the relation E = E((Xn, ψn)n∈N) on ∏n∈N Xn by (x, y) ∈ E((Xn, ψn)n∈N) <=>Σn∈Nψn(x(n), y(n)) < +∞. If E is an equiv- alence relation and all ψn, n ∈ N, are Borel, we show a trichotomy that either IRN/e1≤B E, E1≤B E, or E≤B E0. We also prove that, for a rather general case, E((Xn, ψn)n∈N) is an equivalence relation iff it is an ep-like equivalence relation.Let Xn, n ∈ N be a sequence of non-empty sets, ψn ： Xn2 → IR＋. We consider the relation E = E（（Xn, ψn）n∈N） on ∏n∈N Xn by （x, y） ∈ E（（Xn, ψn）n∈N） 〈=〉Σn∈Nψn（x（n）, y（n）） 〈 ＋∞. If E is an equiv- alence relation and all ψn, n ∈ N, are Borel, we show a trichotomy that either IRN/e1≤B E, E1≤B E, or E≤B E0. We also prove that, for a rather general case, E（（Xn, ψn）n∈N） is an equivalence relation iff it is an ep-like equivalence relation.

关 键 词：Borel reducibility equivalence relation METRIZATION 

分 类 号：O153.3[理学—数学] TP24[理学—基础数学]