### Initially $\kappa $-compact spaces for large $\kappa $

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 $p\in X$.