On non-measurability of ${\ell _\infty }/c_0$ in its second dual
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- by Dennis K. Burke and Roman Pol PDF
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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 Čech-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’skiĭ.References
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Additional Information
- Dennis K. Burke
- Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056-9987
- Email: burkedk@muohio.edu
- Roman Pol
- Affiliation: Faculty of Mathematics, Informatics and Mechanics, Warsaw University, 00-927 Warsaw, Poland
- Email: pol@mimuw.edu.pl
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3955-3959
- MSC (2000): Primary 54C35; Secondary 28A05, 54H05
- DOI: https://doi.org/10.1090/S0002-9939-03-06983-1
- MathSciNet review: 1999946