Cardinal inequalities involving the Hausdorff pseudocharacter

Abstract

AbstractWe establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter $$H\psi (X)$$ H ψ ( X ) . This invariant has the property $$\psi _c(X)\le H\psi (X)\le \chi (X)$$ ψ c ( X ) ≤ H ψ ( X ) ≤ χ ( X ) for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by $$2^{pwL_c(X)H\psi (X)}$$ 2 p w L c ( X ) H ψ ( X ) , where $$pwL_c(X)\le L(X)$$ p w L c ( X ) ≤ L ( X ) and $$pwL_c(X)\le c(X)$$ p w L 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 $$H\psi (X)=\omega $$ H ψ ( X ) = ω , then $$|X|\le 2^\omega $$ | X | ≤ 2 ω , under the assumption that either $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ 2 < c = c or $${\mathfrak {c}}<\aleph _\omega $$ 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 $$G^{\kappa }_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 κ ( O , O D ) , $$H \psi (X) < \kappa $$ H ψ ( X ) < κ and $$\chi (X) \le 2^{<\kappa }$$ χ ( X ) ≤ 2 < κ , then $$|X| \le 2^{<\kappa }$$ | 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 $$H\psi (X)=\omega $$ H ψ ( X ) = ω , then $$|X|\le 2^\omega $$ | X | ≤ 2 ω under the assumption $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ 2 < c = c . Finally, we show $$|X|\le 2^{wL(X)H\psi (X)}$$ | X | ≤ 2 w L ( X ) H ψ ( X ) if X is a Hausdorff space with a $$\pi $$ π -base with elements with compact closures. This is a variation of a result of Bella, Carlson, and Gotchev.