$\sigma$-homogeneity of Borel sets

Abstract

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

4 pages