Ideal approach to convergence in functional spaces

Serhii Bardyla, Jaroslav Supina, Lyubomyr Zdomskyy

Published 2021-11-09, updated 2023-01-18Version 2

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy, which was later reposed by A. Miller, T. Orenshtein and B. Tsaban. Namely, we show that under p = c there is a \delta-set, which is not a \gamma-set. Thus we construct a set of reals A such that although Cp(A), the space of all real-valued continuous functions on A, does not have Frechet-Urysohn property, Cp(A) still possesses Pytkeev property. Moreover, under CH we construct a \pi-set which is not a \delta-set solving a problem by M. Sakai. In fact, we construct various examples of \delta-sets, which are not \gamma-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of Frechet-Urysohn property for many different Borel ideals in the realm of functional spaces.