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Spaces on which every pointwise convergent series of continuous functions converges pseudo-normally


Authors: Lev Bukovsky and Krzysztof Ciesielski
Journal: Proc. Amer. Math. Soc. 133 (2005), 605-611
MSC (2000): Primary 54G99, 03E35; Secondary 54A35, 54C30
DOI: https://doi.org/10.1090/S0002-9939-04-07376-9
Published electronically: August 25, 2004
MathSciNet review: 2093085
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Abstract: A topological space $X$ is a $\Sigma\Sigma^*$-space provided that, for every sequence $\langle f_n\rangle_{n=0}^\infty$ of continuous functions from $X$ to $\mathbb{R} $, if the series $\sum_{n=0}^\infty\vert f_n\vert$ converges pointwise, then it converges pseudo-normally. We show that every regular Lindelöf $\Sigma\Sigma^*$-space has the Rothberger property. We also construct, under the continuum hypothesis, a $\Sigma\Sigma^*$-subset of $\mathbb{R} $ of cardinality continuum.


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

  • 1. T. Bartoszynski, H. Judah, Set Theory, A K Peters, 1995. MR 1350295 (96k:03002)
  • 2. L. Bukovský, I. Rec\law, and M. Repický, Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology Appl. 41 (1991), 25-40. MR 1129696 (93b:54037)
  • 3. L. Bukovský, I. Rec\law, and M. Repický, Spaces not distinguishing convergences of real-valued functions, Topology Appl. 112 (2001), 13-40. MR 1815270 (2002e:54010)
  • 4. K. Ciesielski, Set Theory for the Working Mathematician, London Math. Soc. Stud. Texts 39, Cambridge Univ. Press, 1997. MR 1475462 (99c:04001)
  • 5. F. Galvin, A.W. Miller, $\gamma$-sets and other singular sets of real numbers, Topology Appl. 17 (1984), 145-155. MR 0738943 (85f:54011)
  • 6. J. Gerlits, Zs. Nagy, Some properties of $C(X)$, I, Topology Appl. 14 (1982), 151-161. MR 0667661 (84f:54021)
  • 7. W. Just, A.W. Miller, M. Scheepers, and P.J. Szeptycki, The combinatorics of open covers II, Topology Appl. 73 (1996), 241-266. MR 1419798 (98g:03115a)
  • 8. A.W. Miller, Special Subsets of the Real Line, in: Handbook of Set-Theoretic Topology (K. Kunen and J.E. Vaughan, eds.), North-Holland (1984), 201-233. MR 0776624 (86i:54037)
  • 9. M. Repický, Porous sets and additivity of Lebesgue measure, Real Anal. Exchange 15(1) (1989/90), 282-298. MR 1042544 (91a:03098)
  • 10. W. Rudin, Real and Complex Analysis, McGraw-Hill Book Company, 1987. MR 0210528 (35:1420)

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Additional Information

Lev Bukovsky
Affiliation: Institute of Mathematics, Faculty of Sciences, P. J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
Email: bukovsky@kosice.upjs.sk

Krzysztof Ciesielski
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
Email: K_Cies@math.wvu.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07376-9
Keywords: $\Sigma\Sigma^*$-space, Rothberger property, quasinormal convergence, pseudo-normal convergence
Received by editor(s): January 8, 2003
Received by editor(s) in revised form: June 5, 2003
Published electronically: August 25, 2004
Additional Notes: This work was partially supported by NATO Grant PST.CLG.977652. The second author was also supported by 2002/03 West Virginia University Senate Research Grant.
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society

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