Point countability in generalized ordered spaces

In this paper we introduce a property that is a necessary and sufficient condition for a generalized ordered space X with a point-countable base to have a σ-disjoint base. The property is that there are subsets U(n) and D(n) of X such that U(n) is open in X and D(n) is a discrete-in-itself, relatively closed subset of U(n) such that if p is a point of an open set G, then for some n, we have p ϵ U(n) and D(n) ∩ G ≠ φ. This property is hereditary in a generalized ordered space X and implies hereditary paracompactness of X. We give examples to show that our results are the sharpest possible in ordered spaces and describe the role of Property III in general spaces.