Upper bounds for the tightness of the Gδ -topology

We prove that if X is a regular space with no uncountable free sequences, then the tightness of its Gδ topology is at most the continuum and if X is, in addition, assumed to be Lindelöf then its Gδ topology contains no free sequences of length larger then the continuum. We also show that, surprisingly, the higher cardinal generalization of our theorem does not hold, by constructing a regular space with no free sequences of length larger than ω1, but whose Gδ topology can have arbitrarily large tightness.