Free sequences and the tightness of pseudoradial spaces

Let $F(X)$ be the supremum of cardinalities of free sequences in $X$. We prove that the radial character of every Lindel\"of Hausdorff almost radial space $X$ and the set-tightness of every Lindel\"of Hausdorff space are always bounded above by $F(X)$. Solving a question of Bella, we exhibit a Hausdorff radial space $X$ whose radial character is strictly larger than $F(X)$. We then improve a result of Dow, Juh\'asz, Soukup, Szentmikl\'ossy and Weiss by proving that if $X$ is a Lindel\"of Hausdorff space, and $X_\delta$ denotes the $G_\delta$ topology on $X$ then $t(X_\delta) \leq 2^{t(X)}$. Finally, we exploit this to prove that if $X$ is a Lindel\"of Hausdorff pseudoradial space then $F(X_\delta) \leq 2^{F(X)}$, which partially answer a question of Bella and ourselves.