A non-metrizable compact linearly ordered topological space, every subspace of which has a 𝜎-minimal base

Abstract

A collection D \mathcal {D} of subsets of a space is minimal if each element of D \mathcal {D} contains a point which is not contained in any other element of D \mathcal {D} . A base of a topological space is σ \sigma -minimal if it can be written as a union of countably many minimal collections. We will construct a compact linearly ordered space X X satisfying that X X is not metrizable and every subspace of X X has a σ \sigma -minimal base for its relative topology. This answers a problem of Bennett and Lutzer in the negative.