TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

Abstract. An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolk for characterizing Cech-complete spaces. We also show that the absolute Borel class of X is determined by the uniform structure of the space of continuous functions Cp(X); however the case of absolute Gδ metric spaces is still open. More precisely, we prove that, for metrizable spaces X and Y, if : Cp(X) → Cp(Y) is a uniformly continuous surjection and X is an absolute Borel set of multiplicative (resp., additive) class α, α> 1, then Y is also an absolute Borel set of the same class. This result is new even if is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the Cech-completeness of a metric space X is determined by the linear structure of Cp(X). 1