Smoothness on Products of Perfect Sets

Definitions and mathematical statement

Let $\omega=\{0,1,2,\ldots\}$ and equip $X={\mathbb{R}}^\omega$ with its product topology and Borel structure. A Borel equivalence relation on $X$ is an equivalence relation $E\subseteq X\times X$ that is Borel. A nonempty perfect set in ${\mathbb{R}}$ is a nonempty closed set with no isolated points.

For equivalence relations $E$ on $X$ and $F$ on a standard Borel space $Y$, write $E\leq_B F$ if there is a Borel map $r:X\to Y$ such that \[ \forall x,x'\in X,\qquad x\mathrel E x'\ \Longleftrightarrow\ r(x)\mathrel F r(x'). \] For a Borel action of a Polish group $G$ on a standard Borel space $Y$, its orbit equivalence relation is \[ y\mathrel{E_G^Y}y'\quad\Longleftrightarrow\quad (\exists g\in G)\;g\cdot y=y'. \] Here a Polish group is a topological group whose topology is separable and completely metrizable; a standard Borel space is a measurable space isomorphic to the Borel space of a Polish space. The orbit relation is not assumed to be Borel merely because the action is Borel.

Question under these conventions. For every Borel equivalence relation $E$ on ${\mathbb{R}}^\omega$ such that $E\leq_B E_G^Y$ for some such action, does there exist a sequence $(P_n)_{n\in\omega}$ of nonempty perfect subsets of ${\mathbb{R}}$ such that, with \[ P=\prod_{n\in\omega}P_n, \] the restricted relation $E\mathbin{\upharpoonright}P=E\cap(P\times P)$ is smooth? Explicitly, the requested conclusion is the existence of a Borel map $f:P\to{\mathbb{R}}$, for the relative Borel structure on $P$, satisfying \[ \forall x,x'\in P,\qquad x\mathrel E x'\ \Longleftrightarrow\ f(x)=f(x'). \] The supplied statement does not explicitly specify the reducibility notion or the class of group actions meant by "orbit equivalence." The formulation above uses Borel reducibility and Polish group actions as a conventional interpretation requiring confirmation. It interprets "a product of perfect sets" as a full product of nonempty perfect coordinate sets; nonemptiness excludes a vacuous conclusion. These conventions are not explicitly defined in the supplied source excerpt.

Short English statement

Under the stated descriptive set theory conventions, must every Borel equivalence relation on ${\mathbb{R}}^\omega$ that is Borel reducible to an orbit equivalence relation admit a product of nonempty perfect coordinate sets on which equivalence is classified by equality of a Borel real-valued invariant? The supplied record labels this problem open; that status has not been independently verified.

Sources

• Supplied local UnsolvedMath problem registry record: permanent ID $30001678$, title "Smoothness on Products of Perfect Sets," collection "Oberwolfach Reports Open Problems," exact source problem code catalogue entry. The proposed year is recorded as 2011; no individual proposer is identified.
• Supplied source citation and literature link labeled "Oberwolfach problem statement": Set Theory (2011), Oberwolfach Reports, pp. 85--140, DOI 10.4171/owr/2011/02. https://doi.org/10.4171/owr/2011/02. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature link labeled "canonization on perfect cubes": https://arxiv.org/abs/1804.05174. No authors, precise result, or applicability hypotheses are supplied for this reference.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not an independently verified assessment. The record labels the problem open and reports that product-perfect-set canonization is known in special settings while a general smoothness assertion for an arbitrary Borel orbit equivalence relation remains unresolved. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying verification note states only that TeX notation was enclosed in math delimiters without changing mathematical content; it does not establish mathematical correctness or current research status.