The effect of forcing axioms on the tightness of the $G_δ$ modification

Abstract

We show that $\mathsf{PFA}$ implies that the tightness $t(X_δ)$ of the $G_δ$-modification of a Fréchet $α_1$-space $X$ is at most $ω_1$, while $\Box(κ)$ implies that there is a Fréchet $α_1$-space with $G_δ$-tightness equal to $κ$. We use the example constructed from $\Box(κ)$ to show that a local version of the bound $t(X_δ)\le 2^{t(X)L(X)}$ does not hold. We also construct, assuming $\mathsf{MA}$, an example of a Fréchet space whose $G_δ$-tightness is larger than $ω_1$.