Weak $P$-points for regulars

The main result of this paper is to show that for any compact space $X$ and any family $\{f_{\alpha }\text{\hspace{0.17em}⁣}:\alpha <2^{\kappa },f_{\alpha }\text{\hspace{0.17em}⁣}:X\beta \kappa \}$ 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 ${\kappa }^{\ast }$, whenever $\kappa$ is regular.