On the Orderability Problem and the Interval Topology

TL;DR

This paper explores the role of the interval topology in characterizing linearly ordered topological spaces and investigates extending these concepts to more general transitive relations beyond linear orders.

Contribution

It analyzes the interval topology's role in the Orderability Problem and proposes extensions to non-linear transitive relations.

Findings

Interval topology is crucial in characterizing LOTS.

Extensions to non-linear transitive relations are examined.

Provides insights into the structure of ordered topological spaces.

Abstract

The class of LOTS (linearly ordered topological spaces, i.e. spaces equipped with a topology generated by a linear order) contains many important spaces, like the set of real numbers, the set of rational numbers and the ordinals. Such spaces have rich topological properties, which are not necessarily hereditary. The Orderability Problem, a very important question on whether a topological space admits a linear order which generates a topology equal to the topology of the space, was given a general solution by J. van Dalen and E. Wattel, in 1973. In this article we first investigate the role of the interval topology in van Dalen's and Wattel's characterization of LOTS, and we then examine ways to extend this model to transitive relations that are not necessarily linear orders.