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

Abstract

We show that $\mathsf{PFA}$ implies that the tightness $t(X_{\delta })$ of the $G_{\delta }$-modification of a Fréchet ${\alpha }_1$-space $X$ is at most ${\omega }_1$, while $\square (\kappa )$ implies that there is a Fréchet ${\alpha }_1$-space with $G_{\delta }$-tightness equal to $\kappa$. We use the example constructed from $\square (\kappa )$ 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 Fréchet space whose $G_{\delta }$-tightness is larger than ${\omega }_1$.