Cardinal inequalities involving the Hausdorff pseudocharacter

We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter Hψ(X) . This invariant has the property ψ_c(X) ≤ Hψ(X) ≤ χ(X) for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by 2pwLc(X)Hψ(X) , where pwL_c(X) ≤ L(X) and pwL_c(X) ≤ c(X) . This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if X is a Hausdorff linearly Lindelöf space such that Hψ(X) = ω , then | X| ≤ 2 ^ω , under the assumption that either 2 ^<c= c or c< ℵ_ω . The following game-theoretic result is shown: if X is a regular space such that player two has a winning strategy in the game G1κ(O,OD) , Hψ(X) < κ and χ(X) ≤ 2 ^<κ , then | X| ≤ 2 ^<κ . This improves a result of Aurichi, Bella, and Spadaro. Generalizing a result for first-countable spaces, we demonstrate that if X is a Hausdorff almost discretely Lindelöf space satisfying Hψ(X) = ω , then | X| ≤ 2 ^ω under the assumption 2 ^<c= c . Finally, we show | X| ≤ 2 ^wL(X)Hψ(X) if X is a Hausdorff space with a π -base with elements with compact closures. This is a variation of a result of Bella, Carlson, and Gotchev.