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

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

2010-06-23

arxiv.org/abs/1006.4405v1

Mathematics

Logic

8 pages, submitted

Scientific paper

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.

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