Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2512.17314

Circular orders: topology and continuous actions

Michael Megrelishvili
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GNmath.DSmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael Megrelishvili. 2026-04-24. Circular orders: topology and continuous actions. https://arxiv.org/abs/2512.17314

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Three Problems on Separable Quotients of Precompact Abelian Groups

We address three problems on separable quotients of topological groups posed by Leiderman, Morris, and Tkachenko in \cite{LMT} published on Israel Journal of Mathematics. First, we construct in ZFC a connected Baire Pontryagin-reflexive dense subgroup of $\T^{\cc}$ whose countable subgroups are $h$-embedded and whose uncountable subgroups are dense. Its underlying abstract group is the circle group, and all its compact subsets are finite. Second, we construct a zero-dimensional Baire Pontryagin-reflexive example with the same subgroup properties whose underlying group is free abelian of rank $\cc$. Both examples have no nontrivial separable Hausdorff quotient. Third, for the group constructed in their Theorem~3.5, we determine every closed subgroup of every finite power up to an integral change of coordinates and prove that every countable subgroup of every Hausdorff quotient of a finite power is $h$-embedded and closed. The same conclusions hold for our free Baire reflexive example. These results answer Problem~1.25 negatively, Problem~3.12 affirmatively and realize all three regularity properties in Problem~3.14 simultaneously in \cite{LMT}.

math.GN↗
arXiv2026

Topological Vector Group Topologies Between the Minimal Topology and the Usual Topology on the Real Line

For every positive sequence that tends to zero faster than every fixed exponential, we construct a Hausdorff topological Vector Group topology on the additive group of real numbers. It lies strictly between the minimal Hausdorff topological Vector Group topology and the usual topology. The construction is illustrated by factorial powers, quadratic exponential decay, and prime radicals divided by a quadratic exponential. We also record a finite scalar covering criterion for comparing two such topologies.

math.GN↗
arXiv2026

Sobriety of Scott topologies under countability conditions

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

math.GN↗