On the tightness of $${G_\delta}$$ G δ -modifications

Abstract

The $${G_\delta}$$ -modification $${X_\delta}$$ 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 $${G_\delta}$$ subsets of X. Bella and Spadaro recently asked the following question(s): Is $${t(X_\delta) \le 2^{t(X)}}$$ true for every (compact) T2 space X? In this note we answer both questions: In the compact case affirmatively and in the non-compact case negatively. In the latter case we even show that it is consistent with ZFC that no upper bound exists for the tightness of the $${G_\delta}$$ -modifications of countably tight, even Frechet spaces.