2023/07/29 by Leiderman, Arkady, Szeptycki, Paul
#46A03 #54C35 #54G12 #54H05 #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2307.16047
Δ-spaces have been defined by a natural generalization of a classical notion of Δ-sets of reals to Tychonoff topological spaces; moreover, the class Δ of all Δ-spaces consists precisely of those X for which the locally convex space Cp(X) is distinguished. The aim of this article is to better understand the boundaries of the class Δ, by presenting new examples and counter-examples. 1) We examine when trees considered as topological spaces equipped with the interval topology belong to Δ. In particular, we prove that no Souslin tree is a Δ-space. Other main results are connected with the study of 2) Ψ-spaces built on maximal almost disjoint families of countable sets; and 3) Ladder system spaces. It is consistent with CH that all ladder system spaces on ω1 are in Δ. We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on ω1 which is not in Δ. We resolve several open problems posed in the literature.