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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Countable network weight
and multiplication of Borel sets


Authors: D. H. Fremlin, R. A. Johnson and E. Wajch
Journal: Proc. Amer. Math. Soc. 124 (1996), 2897-2903
MSC (1991): Primary 54H05, 28A05
MathSciNet review: 1343692
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Abstract: A space $X$ Borel multiplies with a space $Y$ if each Borel set of $X\times Y$ is a member of the $\sigma $-algebra in $X\times Y$ generated by Borel rectangles. We show that a regular space $X$ Borel multiplies with every regular space if and only if $X$ has a countable network. We give an example of a Hausdorff space with a countable network which fails to Borel multiply with any non-separable metric space. In passing, we obtain a characterization of those spaces which Borel multiply with the space of countable ordinals, and an internal necessary and sufficient condition for $X$ to Borel multiply with every metric space.


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Additional Information

D. H. Fremlin
Affiliation: Department of Mathematics, Essex University, Colchester C04 3SQ, England
Email: fremdh@essex.ac.uk

R. A. Johnson
Affiliation: Department of Mathematics, Washington State University, Pullman, Washington 99164
Email: johnson@beta.math.wsu.edu

E. Wajch
Affiliation: Institute of Mathematics, University of Łódź, S. Banacha 22, 90-238 Łódź, Poland
Email: ewajch@krysia.uni.lodz.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03488-0
PII: S 0002-9939(96)03488-0
Keywords: Borel set, product $\sigma $-algebra, countable network, hereditary separability, hereditary Lindel\"{o}f property, metric space, countable ordinals
Received by editor(s): February 21, 1995
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1996 American Mathematical Society