Canonization of smooth equivalence relations on infinite-dimensional perfect cubes

Abstract

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

A revised version