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On non-measurability of ${\ell_\infty}/c_0$ in its second dual

Authors: Dennis K. Burke and Roman Pol
Journal: Proc. Amer. Math. Soc. 131 (2003), 3955-3959
MSC (2000): Primary 54C35; Secondary 28A05, 54H05
Published electronically: June 30, 2003
MathSciNet review: 1999946
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Abstract: We show that ${\ell_\infty}/c_0=C(\mathbb{N} ^*)$ with the weak topology is not an intersection of $\aleph_1$ Borel sets in its Cech-Stone extension (and hence in any compactification). Assuming (CH), this implies that $(C(\mathbb{N} ^*),\mathrm{weak})$ has no continuous injection onto a Borel set in a compact space, or onto a Lindelöf space. Under (CH), this answers a question of Arhangel'ski{\u{\i}}\kern.15em.

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

  • [Ar1] A. V. Arhangel'ski{\u{\i}}\kern.15em, $C_p$-theory, Recent Progress in General Topology, Ed. by M. Husek and J. van Mill, Elsevier Science Publishers, 1992.
  • [Ar2] A. V. Arhangel'ski{\u{\i}}\kern.15em, On condensations of $C_p$-spaces onto compacta, Proc. Amer. Math. Soc., 128 (2000) 1881-1883. MR 2001c:54002
  • [AP] A. V. Arhangel'ski{\u{\i}}\kern.15emand O. I. Pavlov, A note on condensation of $C_p(X)$ onto compacta, preprint.
  • [C-S] F. Casarrubias-Segura, On compact weaker topologies in function spaces, Topology Appl., 115 (2001) 291-298. MR 2002g:54019
  • [De] G. Debs, Points de continuité d'une fonction séparément continue. II, Proc. Amer. Math. Soc., 99 (1987) 777-782. MR 88d:54013
  • [DJP] A. Dow, H. Junnila, and J. Pelant, Weak covering properties of weak topologies, Proc. London Math. Soc. (3) 75 (1997) 349-368. MR 98i:54010
  • [DJP2] A. Dow, H. Junnila, and J. Pelant, More on weak covering properties of weak topologies, preprint.
  • [Ed] G. A. Edgar, Measurability in a Banach Space, II, Indiana Univ. Math. J., 28 (1979), no. 4, 559-579. MR 81d:28016
  • [Ha] R. G. Haydon, Boolean rings that are Baire spaces, Serdica Math. J., (2001), no. 2, 91-106.
  • [JNR] J. E. Jayne, I. Namioka, and C. A. Rogers, $\sigma$-fragmentable Banach spaces, Mathematika, 39 (1992), no. 78, 197-215. MR 94c:46028
  • [Ke] A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. MR 96e:03057
  • [Ta] M. Talagrand, Comparaison des boreliens d'un espace de Banach pour les topologies fortes et faibles, Indiana Univ. Math. J. 27 (1978), no. 6., 1001-1004. MR 80d:46033

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Additional Information

Dennis K. Burke
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056-9987

Roman Pol
Affiliation: Faculty of Mathematics, Informatics and Mechanics, Warsaw University, 00-927 Warsaw, Poland

Keywords: Borel sets, C-measurable, function spaces, pointwise topology, weak topology, condensation, \v Cech-Stone compactification
Received by editor(s): July 9, 2002
Received by editor(s) in revised form: August 1, 2002
Published electronically: June 30, 2003
Additional Notes: The second author is grateful to the Department of Mathematics and Statistics at Miami University for its hospitality during the work on this paper
The authors wish to thank the referee for a very prompt report with suggestions which improved the exposition of this paper
Communicated by: Alan Dow
Article copyright: © Copyright 2003 American Mathematical Society

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