FUNCTION SPACES AND THE GERLITS–NAGY δ-PROPERTY

For a set X of real numbers, let C(X) be the space of continuous real-valued functions, with the topology of pointwise convergence. In their seminal paper, Gerlits and Nagy [Topology Appl. 14 (1982), pp. 151–161] introduced their covering property δ, and asked whether, for every δ-set X, the space C(X) is Fréchet–Urysohn. In a recent breakthrough, Bardyla, Šupina and Zdomskyy [Trans. Amer. Math. Soc. 376 (2023), pp. 8495–8528] proved that the Continuum Hypothesis implies a counterexample. Here, we prove that a δ-set need not even have the weak quasinormal convergence property (wQN), a classic property of the space C(X) that is more general than Fréchet–Urysohn. This solves the last standing open problem about potential implications among the δ-property and the covering properties in the Scheepers Diagram. We also establish a stronger form of the main result.