Borel reducibility and finitely Holder(α) embeddability

Abstract

Let $(X_n,d_n),n\in \mathbb{N}$ be a sequence of pseudo-metric spaces, $p\ge 1$. For $x,y\in {\prod }_{n\in \mathbb{N}}{X}_n$, let $(x,y)\in E((X_n{)}_{n\in \mathbb{N}};p)\iff {\sum }_{n\in \mathbb{N}}{d}_n(x(n),y(n){)}^p<+\infty$. For Borel reducibility between equivalence relations $E((X_n{)}_{n\in \mathbb{N}};p)$, we show it is closely related to finitely Hölder($\alpha$) embeddability between pseudo-metric spaces.

18pages, submitted