The comb-like representations of cellular ordinal balleans

Abstract

Given two ordinal λ and γ, let f:[0,λ)→[0,γ) be a function such that, for each α<γ, sup{f(t):t∈[0,α]}<γ. We define a mapping df:[0,λ)×[0,λ)⟶[0,γ) by the rule: if x<y then df(x,y)=df(y,x)=sup{f(t):t∈(x,y]}, d(x,x)=0. The pair ([0,λ),df) is called a γ−comb defined by f. We show that each cellular ordinal ballean can be represented as a γ−comb. In General Asymptology, cellular ordinal balleans play a part of ultrametric spaces.