The effect of forcing axioms on the tightness of the -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.
