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

Let $X$ be a Polish space, $d$ a pseudo-metric on $X$. If $\{(u,v):d(u,v)<δ\}$ is ${\bfΠ}^1_1$ for each $δ>0$, we show that either $(X,d)$ is separable or there are $δ>0$ and a perfect set $C\subseteq X$ such that $d(u,v)\geδ$ 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.

No reviews yet.

No comments yet.