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 $F(X)$. 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 $F(X_delta) leq 2^{F(X)}$.