Abstract:This work presents some cardinal inequalities in which appears the closed pseudo-character, $\psi _c$, of a space. \par Using one of them --- $\psi _c(X) \le 2^{d(X)}$ 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 \leq \kappa $. \par And then, using an idea of A. Dow, we prove that initially $\kappa $-compact $T_2$ spaces are in fact compact for $\kappa = 2^{F(X)}$, $2^{s(X)}$, $2^{t(X)}$, $2^{\chi (X)}$, $2^{\psi _c(X)}$ or $\kappa = \max \{\tau ^+, \tau ^{<\tau }\}$, where $\tau > t(p,X)$ for all $p \in X$.
Keywords: initially $\kappa $-compact space, $\kappa $-bounded space, closed pseudocharacter, cardinal inequalities
AMS Subject Classification: 54D20, 54A25, 54D10