σ-Homogeneity of Borel sets

Abstract

We give an affirmative answer to the following question: Is any Borel subset of a Cantor set 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 ⊂ R n can be partitioned into countably many h-homogeneous subspaces that are G δ -sets in X.We will denote by R , P, Q, and C the spaces of real, irrational, rational numbers, and a Cantor set, respectively.Recall that a zero-dimensional topological space X is h-homogeneous if U is homeomorphic to X for each nonempty clopen subset U ⊂ X. More about topological properties of h-homogeneous spaces see, for example, in [5], [6], [7], [12].2000 Mathematics Subject Classification. Primary 54H05, 03E15. Secondary 03E60, 28A05.

Zero-dimensional $σ$-homogeneous spaces

“…Notice that Theorems 8.2 and 8.3 can be obtained by setting Σ " P in the above result, while [Os,Theorem 1] and [vE2,Theorem 5.1] can be obtained by setting Σ " Borel.…”