On $k$-ranks of topological spaces

TL;DR

This paper introduces the concept of $k$-rank for $T_0$ spaces, establishing its properties and existence for all ordinals, thereby advancing the understanding of well-filtered and related spaces in topology.

Contribution

It defines $k$-rank as a new ordinal measure for $T_0$ spaces and proves the existence of spaces with arbitrary $k$-rank, extending the framework of well-filtered spaces.

Findings

Existence of $k$-well-filtered reflection for any $T_0$ space.

Construction of $T_0$ spaces with arbitrary $k$-rank.

Corollary that $d$-rank and $wf$-rank can also attain any ordinal.

Abstract

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 $\alpha$.