Cardinal inequalities involving the Hausdorff pseudocharacter

Abstract We establish several bounds on the cardinality of a topological space involving Hausdorff pseudocharacter $$H\psi (X)$$ H ψ ( X ) . This invariant has property $$\psi _c(X)\le H\psi (X)\le \chi c ≤ χ for X show is bounded by $$2^{pwL_c(X)H\psi (X)}$$ 2 p w L , where $$pwL_c(X)\le L(X)$$ and c(X)$$ generalizes results Bella Spadaro, as well Hodel. additionally that if linearly Lindelöf such (X)=\omega $$ = ω then $$|X|\le 2^\omega | under assumption either $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ < or $${\mathfrak {c}}<\aleph _\omega ℵ The following game-theoretic result shown: regular player two winning strategy in game $$G^{\kappa }_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 κ O , D $$H \psi (X) < \kappa $$\chi \le 2^{<\kappa }$$ $$|X| improves Aurichi, Bella, Spadaro. Generalizing first-countable spaces, we demonstrate almost discretely satisfying Finally, 2^{wL(X)H\psi with $$\pi π -base elements compact closures. variation Carlson, Gotchev.