Monotone metric spaces (Q1757864)

A metric space \((X,d)\) is monotone if there are a linear order \(\leq \) on \(X\) and \(c\in \mathbb R\) such that \(d(x,y) \leq c\cdot d(x,z)\) whenever \(x<y<z\). Countable unions of monotone metric spaces are called \(\sigma\)-monotone. The authors investigate the topological structure of these spaces, in particular their relations with LOTS and GO spaces. They prove that monotone metric spaces are GO spaces, and \(\sigma\)-monotone metric spaces are 1-dimensional. If each compatible metric on a metrizable space \(X\) is \(\sigma\)-monotone, then \(\dim X = 0\).