The $\wf$-rank of topological spaces

Abstract

The class of well-filtered spaces is one of the important class of topological spaces in domain theory. Recently, the existence of well-filterification of $T_0$-spaces has been established. In this paper, we introduce the definition of $\wf$-rank and prove that for any ordinal $\alpha$, there exists a $T_0$-space whose $\wf$-rank equals  $\alpha$.