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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
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.

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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

Krzysztof Ciesielski
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310

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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