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Keywords:
initially $\kappa$-compact space; $\kappa$-bounded space; closed pseudocharacter; cardinal inequalities
Summary:
This work presents some cardinal inequalities in which appears the closed pseu\-do-\-cha\-racter, $\psi_c$, of a space. 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 $2^\lambda \leq \kappa$. 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 $p \in X$.
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