  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.
