Canonization of Smooth Equivalence Relations on Infinite-Dimensional E0-Large Products

Abstract

We propose a canonization scheme for smooth equivalence relations on ${\mathbb{R}}^{\omega }$ modulo restriction to ${\mathsf{E}}_0$-large infinite products. It shows that, given a pair of Borel smooth equivalence relations $\mathsf{E}$, $\mathsf{F}$ on ${\mathbb{R}}^{\omega }$, there is an infinite ${\mathsf{E}}_0$-large perfect product $P\subseteq {\mathbb{R}}^{\omega }$ such that either $\mathsf{F}\subseteq \mathsf{E}$ on $P$, or, for some $\ell <\omega$, the following is true for all $x,y\in P$: $x\mathsf{E}y$ implies $x\left(\ell \right)=y\left(\ell \right)$, and $x\upharpoonright (\omega \setminus \{\ell \})=y\upharpoonright (\omega \setminus \{\ell \})$ implies $x\mathsf{F}y$.