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). We then improve a result of Dow, Juhász, Soukup, Szentmiklóssy and Weiss by proving that if X is a Lindelöf Hausdorff space, and Xδ denotes the Gδ topology on X then t(Xδ) ≤ 2 t ( X ). Finally, we exploit this to prove that if X is a Lindelöf Hausdorff pseudoradial space then F(Xδ) ≤ 2 F ( X ).