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Absolute Borel sets and function spaces. (English) Zbl 0880.54023
Summary: 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 Frolík for characterizing Čech-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 \(G_{\delta}\) metric spaces is still open. More precisely, we prove that, for metrizable spaces \(X\) and \(Y\), if \(\Phi: C_{p}(X) \rightarrow C_{p}(Y)\) 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 \(\Phi\) is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the Čech-completeness of a metric space \(X\) is determined by the linear structure of \(C_{p}(X)\).

MSC:
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54C35 Function spaces in general topology
03E15 Descriptive set theory
54E35 Metric spaces, metrizability
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