  We prove that if $X$ is a regular space with no uncountable free sequences,
then the tightness of its $G_\delta$ topology is at most continuum and if $X$
is in addition Lindel\"of then its $G_\delta$ 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 $\omega_1$, but whose
$G_\delta$ topology can have arbitrarily large tightness.
