Characterization of $\ell_p$-like and $c_0$-like equivalence relations

TL;DR

This paper investigates the structure of certain equivalence relations derived from pseudo-metrics on Polish spaces, establishing a dichotomy and classifying their complexity within the Borel reducibility hierarchy.

Contribution

It introduces a dichotomy for pseudo-metrics with analytic balls and characterizes the placement of $ ext{ell}_p$-like and $c_0$-like relations in the Borel hierarchy.

Findings

Either the space is separable or contains a perfect set with a uniform distance.

Characterization of $ ext{ell}_p$-like and $c_0$-like relations in the Borel hierarchy.

Provides a dichotomy for pseudo-metrics with analytic balls.

Abstract

Let $X$ be a Polish space, $d$ a pseudo-metric on $X$. If $\{(u,v):d(u,v)<\delta\}$ is ${\bf\Pi}^1_1$ for each $\delta>0$, we show that either $(X,d)$ is separable or there are $\delta>0$ and a perfect set $C\subseteq X$ such that $d(u,v)\ge\delta$ for distinct $u,v\in C$. Granting this dichotomy, we characterize the positions of $\ell_p$-like and $c_0$-like equivalence relations in the Borel reducibility hierarchy.