Upper bounds for the tightness of the $G_{\hat{I}}´$-topology

Abstract

We prove that if $X$ is a regular space with no uncountable free sequences, then the tightness of its $G_{\hat{I}}´$ topology is at most continuum and if $X$ is in addition Lindelöf then its $G_{\hat{I}}´$ topology contains no free sequences of length larger then the continuum. We also show that the higher cardinal generalization of our theorem does not hold, by constructing a regular space with no free sequences of length larger than $\ddot{I}{‰}_1$, but whose $G_{\hat{I}}´$ topology can have arbitrarily large tightness.