Upper bounds for the tightness of the $G_\delta$-topology

TL;DR

This paper investigates bounds on the tightness of the $G_\delta$ topology in regular spaces without uncountable free sequences, establishing continuum bounds under certain conditions and demonstrating limitations of higher cardinal generalizations.

Contribution

It provides new upper bounds for the tightness of the $G_\delta$ topology in specific classes of regular spaces and constructs counterexamples for higher cardinal cases.

Findings

Tightness of $G_\delta$ topology is at most continuum in certain regular spaces.

Lindel"of regular spaces have $G_\delta$ topologies with no free sequences longer than continuum.

Higher cardinal generalizations do not hold universally, as shown by constructed counterexamples.

Abstract

We prove that if $X$ is a regular space with no uncountable free sequences, then the tightness of its $G_\delta$ topology is at most continuum and if $X$ is in addition Lindel\"of then its $G_\delta$ topology contains no free sequences of length larger then the continuum. We also show that the higher cardinal generalization of our theorem does not hold, by constructing a regular space with no free sequences of length larger than $\omega_1$, but whose $G_\delta$ topology can have arbitrarily large tightness.