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Absolute Borel sets and function spaces

Authors: Witold Marciszewski and Jan Pelant
Journal: Trans. Amer. Math. Soc. 349 (1997), 3585-3596
MSC (1991): Primary 04A15, 54H05, 54C35
MathSciNet review: 1401778
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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 Frolík 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 $G_{\delta }$ 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 \v{C}ech-completeness of a metric space $X$ is determined by the linear structure of $C_{p}(X)$.

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Additional Information

Witold Marciszewski
Affiliation: Vrije Universiteit, Faculty of Mathematics and Computer Science, De Boelelaan 1081 a, 1081 HV Amsterdam, The Netherlands
Address at time of publication: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Jan Pelant
Affiliation: Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic

Keywords: Absolute Borel set, function space
Received by editor(s): December 14, 1995
Additional Notes: The first author was supported in part by KBN grant 2 P301 024 07.
The second author was supported in part by the grant GAČR 201/94/0069 and the grant of the Czech Acad. Sci. 119401.
Article copyright: © Copyright 1997 American Mathematical Society

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