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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 $x\restriction{(\omega\smallsetminus\{j\})}=y\restriction{(\omega\smallsetminus\{j\})}$ implies $x\,\mathsf F \,y$., Comment: A revised version

We make use of a finite support product of the Jensen minimal $\varPi^1_2$ singleton forcing to define a model in which $\varPi^1_2$ Uniformization fails for a set with countable cross-sections. We also define appropriate submodels of the same model in which Separation fails for $\varPi^1_3$., Comment: arXiv admin note: text overlap with arXiv:1408.3901

For any abelian Polish sigma-compact group H there exist a sigma-ideal Z over N and a Borel Z-approximate homomorphism f : H --> H^N which is not Z-approximable by a continuous true homomorphism g : H --> H^N.