Ideal approach to convergence in functional spaces

TL;DR

This paper resolves a longstanding open problem by constructing specific sets of reals with unique topological properties in functional spaces, advancing understanding of convergence and set classifications.

Contribution

It constructs examples of elta-sets that are not mma-sets, and explores their properties under various set-theoretic assumptions, addressing multiple open problems.

Findings

Existence of elta-sets that are not mma-sets under p=c.

Construction of a elta-set space that is not Frechet-Urysohn but has the Pytkeev property.

Distinction of ideal variants of the Frechet-Urysohn property for different Borel ideals.

Abstract

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 that is not a \gamma-set. Thus we construct a set of reals A such that the space Cp(A) of all real-valued continuous functions on A is not Frechet-Urysohn, but possesses the Pytkeev property. Moreover, under 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 Frechet-Urysohn property for many different Borel ideals in the realm of functional spaces.