On the tightness of $G_\delta$-modifications

TL;DR

This paper investigates the tightness of $G_\delta$-modifications of topological spaces, proving an upper bound for compact spaces and showing the absence of such bounds in non-compact cases, with consistency results in ZFC.

Contribution

It provides a positive answer for compact spaces and a negative, consistency-based answer for non-compact spaces regarding the tightness bounds of $G_\delta$-modifications.

Findings

$t(X_\delta) \le 2^{t(X)}$ for compact $T_2$ spaces

No universal upper bound exists for non-compact spaces

Consistency results show unbounded tightness in certain non-compact spaces

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 investigated the connection between the values of various cardinal functions taken on $X$ and $X_\delta$, respectively. In their paper, as Question 2, they raised the following problem: Is $t(X_\delta) \le 2^{t(X)}$ 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 $G_\delta$-modifications of countably tight, even Frechet spaces.