Ideal approach to convergence in functional spaces

Abstract

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy [17], which was later reposed by A. Miller, T. Orenshtein and B. Tsaban. Namely, we show that under p = c there is a δ-set, which is not a γ-set. Thus we construct a set of reals A such that although C p (A), the space of all real-valued continuous functions on A, does not have Fréchet-Urysohn property, C p (A) still possesses Pytkeev property. Moreover, under CH we construct a π-set which is not a δ-set solving a problem by M. Sakai. In fact, we construct various examples of δ-sets, which are not γ-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of Fréchet-Urysohn property for many different Borel ideals in the realm of functional spaces.