The effect of forcing axioms on the tightness of the $G_\delta $-modification
Volume 251 / 2020
Fundamenta Mathematicae 251 (2020), 195-202
MSC: Primary 54A20; Secondary 03E57, 03E35.
DOI: 10.4064/fm869-2-2020
Published online: 10 April 2020
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 $-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 Fréchet space whose $G_\delta $-tightness is larger than $\omega _1$.
