Free sequences and the tightness of pseudoradial spaces

Abstract

Let $F(X)$ be the supremum of cardinalities of free sequences in $X$. We prove that the radial character of every Lindelöf Hausdorff almost radial space $X$ and the set-tightness of every Lindelöf 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, Juhász, Soukup, Szentmiklóssy and Weiss by proving that if $X$ is a Lindelöf Hausdorff space, and $X_{\delta }$ denotes the $G_{\delta }$ topology on $X$ then $t(X_{\delta })\le 2^{t(X)}$. Finally, we exploit this to prove that if $X$ is a Lindelöf Hausdorff pseudoradial space then $F(X_{\delta })\le 2^{F(X)}$, which partially answer a question of Bella and ourselves.