  In this paper, the concepts of $K$-subset systems and $k$-well-filtered
spaces are introduced, which provide another uniform approach to $d$-spaces,
$s$-well-filtered spaces (i.e., $\mathcal{U}_{S}$-admissibility) and
well-filtered spaces. We prove that the $k$-well-filtered reflection of any
$T_{0}$ space exists. Meanwhile, we propose the definition of $k$-rank, which
is an ordinal that measures how many steps from a $T_{0}$ space to a
$k$-well-filtered space. Moreover, we derive that for any ordinal $\alpha$,
there exists a $T_{0}$ space whose $k$-rank equals to $\alpha$. One immediate
corollary is that for any ordinal $\alpha$, there exists a $T_{0}$ space whose
$d$-rank (respectively, $wf$-rank) equals to $\alpha$.
