Choban operators in generalized ordered spaces
A Choban operator on a topological space \(X\) is a continuous map from~\(X^2\) to~\(X\) that is constant on the diagonal and that maps each vertical line \(\{x\}\times X\) bijectively onto~\(X\). A topological group has an obvious Choban operator: the map \(\langle a,b\rangle\mapsto ab^{-1}\). The authors investigate this and related notions in the class of GO-spaces. They show that a GO-space with a Choban operator is hereditarily paracompact, that first-countable LOTS with a Choban operator are metrizable and that the real line is the only non-trivial connected LOTS with a Choban operator. Among the delimiting examples are: an ultrapower of the real line as a non-metrizable LOTS that is a topological group; the Sorgenfrey and Michael lines have Choban operators; there are Bernstein subsets of the real line with and without Choban operators; and only \textit{open} intervals of the real line have Choban operators.
scientific article; zbMATH DE number 6145455