Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-96-03488-0
MathSciNet review: 1343692
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. K. Ciesielski & F. Galvin, Cylinder problem, Fund. Math. 127 (1987), 171-176. MR 89e:03079
  • 2. R. Engelking, General Topology, Heldermann, 1989. MR 91c:54001
  • 3. R.J. Gardner & W.F. Pfeffer, Borel measures, Chapt. 22 of [10], 961-1043. MR 86c:28031
  • 4. G. Gruenhage, Generalized metric spaces, Chapt. 10 of [10], 423-501. MR 86h:54038
  • 5. P.R. Halmos, Measure Theory, Van Nostrand, 1950. MR 11:504d
  • 6. R.E. Hodel, On a theorem of Arhangel'skii concerning Lindelöf $p$-spaces, Canad. J. Math. 27 (1975), 459-468. MR 51:11401
  • 7. R.A. Johnson, On product measures and Fubini's theorem in locally compact spaces, Trans. Amer. Math. Soc. 123 (1966), 112-129. MR 33:5832
  • 8. R.A. Johnson, E. Wajch & W. Wilczy\'{n}ski, Metric spaces and multiplication of Borel sets, Rocky Mountain J. Math. 22 (1992), 1341-1347. MR 94c:54072
  • 9. K. Kunen, Inaccessibility properties of cardinals, Ph.D. thesis, Stanford University, Palo Alto, 1968.
  • 10. K. Kunen & J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, North-Holland, 1984. MR 85k:54001
  • 11. J.C. Morgan II, Point Set Theory, Marcel Dekker, 1990. MR 91a:54051
  • 12. B.V. Rao, On discrete Borel spaces and projective sets, Bull. Amer. Math. Soc. 75 (1969), 614-617. MR 39:4014
  • 13. B.V. Rao, On discrete Borel spaces, Acta Math. Sci. Hungaricae 22 (1971), 197-198. MR 44:6503
  • 14. R.H. Sorgenfrey, On the topological product of paracompact spaces, Bull. Amer. Math. Soc. 53 (1947), 631-632. MR 8:594f
  • 15. M. Talagrand, Est-ce que $\ell \sp {\infty }$ est un espace mesurable?, Bull. Sci. Math. 103 (1979), 255-258. MR 80k:03056

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54H05, 28A05

Retrieve articles in all journals with MSC (1991): 54H05, 28A05


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: https://doi.org/10.1090/S0002-9939-96-03488-0
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

American Mathematical Society