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 $p\in X$.

A subset $A$ of a Hausdorff space $X$ is called an $\omega$H-set in $X$ if for every open family $𝒰$ in $X$ such that $A\subset \bigcup 𝒰$ there exists a countable subfamily $𝒱$ of $𝒰$ such that $A\subset \bigcup \{\overline{V}:V\in 𝒱\}$. In this paper we introduce a new cardinal function $t_{s\theta }$ and show that $\left|A\right|\le 2^{t_{s\theta }\left(X\right){\psi }_c\left(X\right)}$ for every $\omega$H-set $A$ of a Hausdorff space $X$.

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $\alpha \le 2^{}$, a topological group G such that $G^{\gamma }$ is countably compact for all cardinals γ < α, but $G^{\alpha }$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M{A}_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M{A}_{countable}$. However, the question has...

We say that a cardinal function $\phi$ reflects an infinite cardinal $\kappa$, if given a topological space $X$ with $\phi \left(X\right)\ge \kappa$, there exists $Y\in {\left[X\right]}^{\le \kappa }$ with $\phi \left(Y\right)\ge \kappa$. We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences...

We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form (∀X)ϕ(X) and (∃X)ϕ(X), for ϕ positive in X and containing no set-quantifiers, when the set variables range over large (= cofinal) subsets of the cardinals. This strengthens the result of Doner-Mostowski-Tarski [3] that (κ,∈), (λ,∈) are elementarily equivalent when κ, λ are uncountable. It follows that we can consistently postulate that the structures $(2^{\kappa },{\left[2^{\kappa }\right]}^{>\kappa },<)$, $(2^{\lambda },{\left[2^{\lambda }\right]}^{>\lambda },<)$ are...