On the Orderability Problem and the Interval Topology

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.

arXiv commentChapter in the Volume Topics in Mathematical Analysis and Applications, in the Optimization and Its Applications Springer Series, T. Rassias and L. Toth Eds, Springer Verlag, 2014. arXiv admin note: text overlap with arXiv:1306.2146

Cited by in corpus (9)

• The causal order on the ambient boundary
• On Two Topologies that were suggested by Zeeman
• The Order on the Light Cone and its induced Topology
• Spacetime Singularities vs. Topologies of Zeeman-Göbel Class
• Nests, and their Role in the Orderability Problem
• On the Possibility of Singularities on the Ambient Boundary
• Spacetimes as topological spaces, and the need to take methods of general topology more seriously
• On the Causal and Topological Structure of the $2$-Dimensional Minkowski Space
• Natural vs. Artificial Topologies on a Relativistic Spacetime