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Canonization of smooth equivalence relations on infinite-dimensional \(\mathsf{E}_0\)-large products. (English) Zbl 1471.03072

Recall that the equivalence relation \(E_0\) on \(2^\omega\) is defined as \[xE_0y\iff\exists m\forall n>m(x(n)=y(n))\] for all \(x,y\in 2^\omega\). An equivalence relation \(E\) on a standard Borel space \(X\) is smooth if there exsits aBorel map \(f:X\to 2^\omega\) satisfying \[xEy\iff f(x)=f(y).\] A Borel set \(X\subseteq 2^\omega\) is \(E_0\)-large if \(E_0\upharpoonright X\) is not smooth.
An infinite perfect product is a set \(P\subseteq(2^\omega)^\omega\) such that \(P=\prod_{l<\omega}P(l)\), where \(P(l)\) is a perfect subset of \(2^\omega\). Furthermore, if each \(P(l)\) is an \(E_0\)-large set, we say that \(P\) is an \(E_0\)-large perfect product.
The main result of this article is:
Theorem. If \(E,F\) are smooth equivalence relations on \((2^\omega)^\omega\), then there is an \(E_0\)-large perfect product \(P\subseteq(2^\omega)^\omega\) such that either \(F\subseteq E\) on \(P\), or, for some \(l<\omega\) and for all \(x,y\in P\), \(xEy\) implies \(x(l)=y(l)\), and \(x\upharpoonright(\omega\setminus\{l\})=y\upharpoonright(\omega\setminus\{l\})\) implies \(xFy\).

MSC:

03E15 Descriptive set theory
03E35 Consistency and independence results
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References:

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