Cardinal inequalities involving the Hausdorff pseudocharacter

Abstract

We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter Hpsi (X). This invariant has the property psi _c(X)le Hpsi (X)le chi (X) for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by 2^{pwL_c(X)Hpsi (X)}, where pwL_c(X)le L(X) and pwL_c(X)le 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 Hpsi (X)=omega , then |X|le 2^omega , under the assumption that either 2^{<{mathfrak {c}}}={mathfrak {c}} or {mathfrak {c}}<aleph _omega . The following game-theoretic result is shown: if X is a regular space such that player two has a winning strategy in the game G^{kappa }_1({mathcal {O}}, {mathcal {O}}_D), H psi (X) < kappa and chi (X) le 2^{<kappa }, then |X| le 2^{<kappa }. 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 Hpsi (X)=omega , then |X|le 2^omega under the assumption 2^{<{mathfrak {c}}}={mathfrak {c}}. Finally, we show |X|le 2^{wL(X)Hpsi (X)} if X is a Hausdorff space with a pi -base with elements with compact closures. This is a variation of a result of Bella, Carlson, and Gotchev.