ABSTRACT
A Borel space is called countably separated if there are countably many Borel sets in the space which separate the points of the space, and countably generated if there are countably many Borel sets in the space which both separate the points of the space and generate the Borel structure. A topological space is called polish if it is separable and if its topology is generated by a complete metric. Hausdorff spaces has a closed range and is a homeomorphism of its domain onto its range. After these theorems on Borel spaces the final result of this section is perhaps somewhat of a let down: it implies that standard Borel spaces are classified up to Borel isomorphism by their cardinality and that every uncountable standard Borel space is Borel isomorphic to [0,1].
