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 $X$.

We call a topological space $\kappa$-compact if every subset of size $\kappa$ has a complete accumulation point in it. Let $\Phi (\mu ,\kappa ,\lambda )$ denote the following statement: $\mu <\kappa <\lambda =cf\left(\lambda \right)$ and there is $\{S_{\xi }:\xi <\lambda \}\subset {\left[\kappa \right]}^{\mu }$ such that $\left|\right\{\xi :|S_{\xi }\cap A|=\mu \left\}\right|<\lambda$ whenever $A\in {\left[\kappa \right]}^{<\kappa }$. We show that if $\Phi (\mu ,\kappa ,\lambda )$ holds and the space $X$ is both $\mu$-compact and $\lambda$-compact then $X$ is $\kappa$-compact as well. Moreover, from PCF theory we deduce $\Phi (cf\left(\kappa \right),\kappa ,{\kappa }^{+})$ for every singular cardinal $\kappa$. As a corollary we get that a linearly Lindelöf and ${\aleph }_{\omega }$-compact space is uncountably compact, that is $\kappa$-compact for all uncountable cardinals...

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...