Upper bounds for the tightness of the $G_δ$-topology

We prove that if $X$ is a regular space with no uncountable free sequences, then the tightness of its $G_δ$ topology is at most continuum and if $X$ is in addition Lindelöf then its $G_δ$ 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 $ω_1$, but whose $G_δ$ topology can have arbitrarily large tightness.

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