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 $\Box (\kappa )$ implies that there is a Fréchet $\alpha _1$-space with $G_\delta $-tig