Cardinal inequalities involving the Hausdorff pseudocharacter

We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter $H\psi(X)$. This invariant has the property $\psi_c(X)\leq H\psi(X)\leq\chi(X)$ for a Hausdorff space $X$. We show the cardinality of a Hausdorff space $X$ is bounded by $2^{pwL_c(X)H\psi(X)}$, where $pwL_c(X)\leq L(X)$ and $pwL_c(X)\leq c(X)$. This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if $X$ is a Hausdorff linearly Lindel\"of space such that $H\psi(X)=\omega$, 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 $\chi(X) \leq 2^{<\kappa}$, then $|X| \leq 2^{<\kappa}$. 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 Lindel\" of space satisfying $H\psi(X)=\omega$, then $|X|\le 2^\omega$ under the assumption $2^{<\mathfrak c}=\mathfrak c$. Finally, we show $|X|\leq 2^{wL(X)H\psi(X)}$ 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.