The comb-like representations of cellular ordinal balleans

Abstract

Given two ordinal $\lambda$ and $\gamma$, let $f:[0,\lambda )\to [0,\gamma )$ be a function such that, for each $\alpha <\gamma$, $\sup \{f(t):t\in [0,\alpha ]\}<\gamma .$ We define a mapping $d_f:[0,\lambda )\times [0,\lambda )⟶[0,\gamma )$ by the rule: if $x<y$ then $d_f(x,y)=d_f(y,x)=\sup \{f(t):t\in (x,y]\}$, $d(x,x)=0$. The pair $([0,\lambda ),d_f)$ is called a $\gamma -$comb defined by $f$. We show that each cellular ordinal ballean can be represented as a $\gamma -$comb. In {\it General Asymptology}, cellular ordinal balleans play a part of ultrametric spaces.

5 pages, preprint