Absolute Borel sets and function spaces

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 Frolik 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 C-p(X); however the case of absolute Ga metric spaces is still open. More precisely, we prove that, far metrizable spaces X and Y, if Phi : C-p(X) --> C-p(Y) is a uniformly continuous surjection and X is an absolute Borel set of multiplicative (resp., additive) class alpha, alpha > 1, then Y is also an absolute Borel set of the same class. This result is new even if cp 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 C-p(X).