Circular orders: topology and continuous actions

TL;DR

This paper develops a comprehensive topological framework for circularly ordered sets, introducing new results and directions, especially in group actions, compactifications, and functions of bounded variation, with applications in topological dynamics.

Contribution

It provides a systematic topological theory of circularly ordered spaces, including compactifications, uniform structures, and generalizations of Helly's theorem, advancing the understanding of circular order topology.

Findings

Convex uniform structure description of circularly ordered compactifications

Topological analysis of Novak's regular completion and its uniformity

Generalizations of Helly's selection theorem for circular orders

Abstract

We study the topology of circularly ordered sets. While the algebraic notion is classical, the general topological theory has received comparatively little attention. In this work we provide a self-contained topological exposition and present several new directions and results. Our aim is to initiate a systematic study of generalized circularly ordered topological spaces and of continuous group actions on them. Provide a convex uniform structure description of circularly ordered compactifications. This yields a topological analysis of Novak's regular completion and its uniformity. Demonstrate that this uniform-structure approach yields several new results in the theory of $G$-compactifications for topological group actions on abstract ordered spaces. Reexamine functions of bounded variation on circularly ordered sets and prove generalizations of Helly's selection theorem (for circular and linear orders). These developments and the systematic analysis of circular order topologies are motivated by recent applications in topological dynamics, particularly in joint works with Eli Glasner, which demonstrate that circularly ordered dynamical systems provide a natural class of tame dynamics.