2018/05/06 by Dow, Alan, Juhász, István, Soukup, Lajos +2
#03E35 #54A25 #54A35 #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.1805.02228
The Gδ-modification Xδ of a topological space X is the space on the same underlying set generated by, i.e. having as a basis, the collection of all Gδ subsets of X. Bella and Spadaro recently investigated the connection between the values of various cardinal functions taken on X and Xδ, respectively. In their paper, as Question 2, they raised the following problem: Is t(Xδ) ≤ 2t(X) true for every (compact) T2 space X? Note that this is actually two questions. In this note we answer both questions: In the compact case affirmatively and in the non-compact case negatively. In fact, in the latter case we even show that it is consistent with ZFC that no upper bound exists for the tightness of the Gδ-modifications of countably tight, even Frechet spaces.