$sigma$-homogeneity of Borel sets

arXiv:1102.3252v1 [math.LO] 16 Feb 2011 σ-HOMOGENEITY OF BOREL SETS ALEXEY OSTROVSKY 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 ⊂ Rn can be partitioned into countably many h-homogeneous subspaces that are Gδ-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]. We call a zero-dimensional metric space X σ-homogeneous if it is a count- able union of h-homogeneous subspaces Xi that are closed in X. It is easily seen that every set Xi \ S j

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