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

Bukovský, Lev; Ciesielski, Krzysztof

doi:10.1090/S0002-9939-04-07376-9

Spaces on which every pointwise convergent series of continuous functions converges pseudo-normally
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by Lev Bukovský and Krzysztof Ciesielski PDF

Proc. Amer. Math. Soc. 133 (2005), 605-611 Request permission

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 $\sum _{n=0}^\infty |f_n|$ 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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