The comb-like representations of cellular ordinal balleans

Abstract

Given two ordinal $λ$ and $γ$, let $f:[0,λ) \rightarrow [0,γ)$ be a function such that, for each $α<γ$, $\sup\{f(t): t\in[0, α]\}<γ.$ We define a mapping $d_{f}: [0,λ)\times [0,λ) \longrightarrow [0,γ)$ 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 $([0,λ), d_{f})$ is called a $γ-$comb defined by $f$. We show that each cellular ordinal ballean can be represented as a $γ-$comb. In {\it General Asymptology}, cellular ordinal balleans play a part of ultrametric spaces.