In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to Fedeli [2] and the third theorem of this paper is a common generalization to: (a) (Arhangel’skii [1]) If $X$ is a $T_1$ space such that (i) $L\left(X\right)t\left(X\right)\le \kappa$, (ii) $\psi \left(X\right)\le 2^{\kappa }$, and (iii) for all $A\in {\left[X\right]}^{\le 2^{\kappa }}$, $\left|\overline{A}\right|\le 2^{\kappa }$, then $\left|X\right|\le 2^{\kappa }$; and (b) (Fedeli [2]) If $X$ is a $T_2$-space then $\left|X\right|\le 2^{aql\left(X\right)t\left(X\right){\psi }_c\left(X\right)}$.

We study maximal almost disjoint (MAD) families of functions in ${\omega }^{\omega }$ that satisfy certain strong combinatorial properties. In particular, we study the notions of strongly and very MAD families of functions. We introduce and study a hierarchy of combinatorial properties lying between strong MADness and very MADness. Proving a conjecture of Brendle, we show that if $cov\left(ℳ\right){<}_{}$, then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from = . Next, we...

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...