# A note on the tightness of $G_\delta$-modifications

## TL;DR

This paper constructs a specific topological space demonstrating the tightness of its $G_\delta$-modification can exceed certain bounds, addressing a question in the field and exploring the influence of set-theoretic assumptions.

## Contribution

It provides a counterexample to a question about the tightness of $G_\delta$-modifications and analyzes how set-theoretic assumptions affect this property.

## Key findings

- Constructed a normal countably tight $T_1$ space with $t(X_\delta) > 2^\omega$.
- Showed that under certain set-theoretic conditions, the tightness can be arbitrarily large up to the least $\omega_1$-strongly compact cardinal.

## Abstract

We construct a normal countably tight $T_1$ space $X$ with $t(X_\delta) >2^\omega$. This is an answer to the question posed by Dow-Juh\'asz-Soukup-Szentmikl\'ossy-Weiss. We also show that if the continuum is not so large, then the tightness of $G_\delta$-modifications of countably tight spaces can be arbitrary large up to the least $\omega_1$-strongly compact cardinal.