Ideal approach to convergence in functional spaces

Bardyla, Serhii; Šupina, Jaroslav; Zdomskyy, Lyubomyr

doi:10.1090/tran/9008

Ideal approach to convergence in functional spaces
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by Serhii Bardyla, Jaroslav Šupina and Lyubomyr Zdomskyy;

Trans. Amer. Math. Soc. 376 (2023), 8495-8528

DOI: https://doi.org/10.1090/tran/9008

Published electronically: September 12, 2023

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Abstract:

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy [Topology Appl. 14 (1982), pp. 151–161], which was later reposed by A. Miller, T. Orenshtein, and B. Tsaban. Namely, we show that under $\mathfrak {p}=\mathfrak {c}$ there is a $\delta$-set that is not a $\gamma$-set. Thus we constructed a subset $A$ of reals such that the space $\mathrm {C}_p(A)$ of all real-valued continuous functions on $A$ is not Fréchet–Urysohn, but possesses the Pytkeev property. Moreover, under $\mathbf {CH}$ we construct a $\pi$-set that is not a $\delta$-set solving a problem by M. Sakai. In fact, we construct various examples of $\delta$-sets that are not $\gamma$-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of the Fréchet–Urysohn property for many different Borel ideals in the realm of functional spaces.

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