Linearly ordered coarse spaces (Q2683646)

A coarse space endowed with a linear order compatible with the coarse structure in the sense of the author [J. Math. Sci., New York 256, No. 6, 779--784 (2021; Zbl 1471.54009)] is said to be linearly ordered. In this paper, the author shows that for a coarse space \((X,\mathcal{E})\) and a linear order \(\leq\) on \(X\), the coarse space \((X, \mathcal{E})\) with \(\leq\) is linearly ordered if and only if \(\mathcal{E}\) has a base \(\mathcal{E}^\prime\) such that \(E^\prime[x]\) is convex for all \(x\in X\) and \(E^\prime\in\mathcal{E}^\prime\) (\(\mathcal{E}^\prime\) is locally convex). He also shows that the asymptotic dimension of a linearly ordered coarse space is either \(0\) or \(1\) and that the family of all right bounded subsets of a linearly ordered metric space has a selector.