On the tightness of $G_{\hat{I}}´$-modifications

Abstract

The $G_{\hat{I}}´$-modification $X_{\hat{I}}´$ 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 $G_{\hat{I}}´$ subsets of $X$. Bella and Spadaro recently investigated the connection between the values of various cardinal functions taken on $X$ and $X_{\hat{I}}´$, respectively. In their paper, as Question 2, they raised the following problem: Is $t(X_{\hat{I}}´)\le 2^{t(X)}$ true for every (compact) $T_2$ 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 $G_{\hat{I}}´$-modifications of countably tight, even Frechet spaces.

9 pages