A trichotomy for a class of equivalence relations

Abstract

Let $X_n,n\in \mathbb{N}$ be a sequence of non-empty sets, ${\psi }_n:X_n^2\to {\mathbb{R}}^{+}$. We consider the relation $E((X_n,{\psi }_n{)}_{n\in \mathbb{N}})$ on ${\prod }_{n\in \mathbb{N}}{X}_n$ by $(x,y)\in E((X_n,{\psi }_n{)}_{n\in \mathbb{N}})\iff {\sum }_{n\in \mathbb{N}}{\psi }_n(x(n),y(n))<+\infty$. If $E((X_n,{\psi }_n{)}_{n\in \mathbb{N}})$ is a Borel equivalence relation, we show a trichotomy that either ${\mathbb{R}}^{\mathbb{N}}/{\ell }_1{\le }_{B}E$, $E_1{\le }_{B}E$, or $E{\le }_B{E}_0$. We also prove that, for a rather general case, $E((X_n,{\psi }_n{)}_{n\in \mathbb{N}})$ is an equivalence relation iff it is an ${\ell }_p$-like equivalence relation.

11 pages. Submitted