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

DOI:
https://doi.org/10.1090/S0002-9947-97-01852-7

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 is determined by the uniform structure of the space of continuous functions ; however the case of absolute metric spaces is still open. More precisely, we prove that, for metrizable spaces and , if is a uniformly continuous surjection and is an absolute Borel set of multiplicative (resp., additive) class , , then 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 \v{C}ech-completeness of a metric space is determined by the linear structure of .

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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

Email:
wmarcisz@cs.vu.nl

**Jan Pelant**

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

Email:
pelant@mbox.cesnet.cz

DOI:
https://doi.org/10.1090/S0002-9947-97-01852-7

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