A metrization theorem for linearly orderable spaces

Abstract

A topological space X is linearly orderable if there is a linear ordering of the set X whose open interval topology coincides with the topology of X. It is known that if a linearly orderable space is semimetrizable then it is, in fact, metrizable [1]. We will use this fact to give a particularly simple metrization theorem for linearly orderable spaces, namely that a linearly orderable space is metrizable if and only if it has a Gs diagonal. This is an interesting analogue of the well-known metrization theorem which' states that a compact Hausdorff space is metrizable if it has a G5 diagonal.