This work presents some cardinal inequalities in which appears the closed pseudo-character, ${\psi }_c$, of a space. Using one of them — ${\psi }_c\left(X\right)\le 2^{d\left(X\right)}$ for $T_2$ spaces — we improve, from $T_3$ to $T_2$ spaces, the well-known result that initially $\kappa$-compact $T_3$ spaces are $\lambda$-bounded for all cardinals $\lambda$ such that $2^{\lambda }\le \kappa$. And then, using an idea of A. Dow, we prove that initially $\kappa$-compact $T_2$ spaces are in fact compact for $\kappa =2^{F\left(X\right)}$, $2^{s\left(X\right)}$, $2^{t\left(X\right)}$, $2^{\chi \left(X\right)}$, $2^{{\psi }_c\left(X\right)}$ or $\kappa =max\{{\tau }^{+},{\tau }^{<\tau }\}$, where $\tau >t(p,X)$ for all $p\in X$.

In this paper two new cardinal functions are introduced and investigated. In particular the following two theorems are proved: (i) If $X$ is a functionally Hausdorff space then $\left|X\right|\le 2^{fs\left(X\right){\psi }_{\tau }\left(X\right)}$; (ii) Let $X$ be a functionally Hausdorff space with $fs\left(X\right)\le \kappa$. Then there is a subset $S$ of $X$ such that $\left|S\right|\le 2^{\kappa }$ and $X=\bigcup \{c{l}_{\tau \theta }\left(A\right):A\in {\left[S\right]}^{\le \kappa }\}$.