$\sigma$-homogeneity of Borel sets

TL;DR

The paper proves that any Borel set in a Cantor set can be decomposed into countably many disjoint $h$-homogeneous subspaces, which are closed in the ambient space, extending to Borel sets in Euclidean spaces.

Contribution

It establishes that all Borel sets in a Cantor set are sums of countably many disjoint $h$-homogeneous closed subspaces, providing a new structural insight.

Findings

Every Borel set in a Cantor set is a sum of disjoint $h$-homogeneous subspaces.

Any Borel subset of $ extbf{R}^n$ can be partitioned into countably many $h$-homogeneous $G_{ ext{delta}}$-sets.

The result affirms a positive answer to a longstanding question about the structure of Borel sets.

Abstract

We give an affirmative answer to the following question: Is any Borel subset of a Cantor set $\textbf{ C}$ a sum of a countable number of pairwise disjoint $h$-homogeneous subspaces that are closed in $X$? It follows that every Borel set $X \subset \textbf{ R}^n$ can be partitioned into countably many $h$-homogeneous subspaces that are $G_{\delta}$-sets in $X$.