The effect of forcing axioms on the tightness of the $G_\delta$ modification

TL;DR

This paper investigates how forcing axioms influence the tightness of the $G_\delta$-modification of certain topological spaces, revealing both bounds under $ extsf{PFA}$ and $ox( ext{kappa})$, and providing counterexamples.

Contribution

It establishes the impact of forcing axioms on $G_\delta$-tightness and constructs examples demonstrating the bounds and limitations of these effects.

Findings

$ extsf{PFA}$ implies $t(X_\delta) extless ext{omega}_1$ for certain spaces.

$ox( ext{kappa})$ implies existence of spaces with $G_\delta$-tightness equal to $ ext{kappa}$.

Counterexamples show local bounds do not always hold, and $ extsf{MA}$ can produce spaces with larger $G_\delta$-tightness.

Abstract

We show that $\mathsf{PFA}$ implies that the tightness $t(X_\delta)$ of the $G_\delta$-modification of a Fr\'echet $\alpha_1$-space $X$ is at most $\omega_1$, while $\Box(\kappa)$ implies that there is a Fr\'echet $\alpha_1$-space with $G_\delta$-tightness equal to $\kappa$. We use the example constructed from $\Box(\kappa)$ to show that a local version of the bound $t(X_\delta)\le 2^{t(X)L(X)}$ does not hold. We also construct, assuming $\mathsf{MA}$, an example of a Fr\'echet space whose $G_\delta$-tightness is larger than $\omega_1$.