I - THE BOREL SUBSETS OF A METRIC SPACE

Abstract

This chapter discusses the study of the properties of Borel subsets of metric spaces. The main result of the isomorphism theorem asserts that Borel subsets of complete separable metric spaces are isomorphic if they have the same cardinality. For any separable metric space X, one writes X0 for the space of all functions x from the positive integers to X. A point x ε X0 can be expressed as a sequence (x1, x2…), where each xn ε X. The same symbol is used to denote the space X as well as the Borel space of which it is the underlying set. The countable product of the space consisting of two points 0 and 1 is denoted by M. M is the space of sequences of 0s and ls. M is a compact metric space in its natural (product) topology.