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 $\left|X\right|\le 2^{aql\left(X\right)t\left(X\right){\psi }_c\left(X\right)}$.

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 $X$.