Weak $P$-points for regulars

The main result of this paper is to show that for any compact space $X$ and any family $\{f_\alpha \colon \alpha< 2^\kappa, f_\alpha \colon X \stackrel{onto}{\longrightarrow} \beta \kappa\}$ of open mappings there exists a point $x \in X$ such that for each $\alpha< 2^\kappa$ either $f_\alpha(x)$ belongs to $\kappa$ or $f_\alpha(x)$ is a weak $P_\kappa$-point in $\kappa^*$, whenever $\kappa$ is regular.