Detecting the real line among one-parametric topological groups
Taras Banakh, Kateryna Makarova, Oles Mazurenko
TL;DR
The paper characterizes the real line within one-parametric topological groups by proving it is the unique metrizable, non-monothetic example. It develops a parametrization framework with uniqueness results and classifies when a one-parametric group is isomorphic to $R$ or to a circle based on kernel structure. Extending to non-metrizable cases, the authors employ Bohr and Borel-Bohr topologies and introduce $ ext{N}$-scaling to relate monotheticity to topological containment. They prove monotheticity for low-weight non-metrizable groups and provide a Bohr-topology–based characterization of monotheticity, including a canonical identification with $R^ lat$ for certain one-parametric groups.
Abstract
We prove that a topological group is isomorphic to the real line if and only if it is a one-parameteric, metrizable, and not monothetic. This result is used in the authors' other paper to prove that one-parametric groups in strictly convex metric group all are topologically isomorphic to the real line. The example of the Bohr topology on the real line demonstrates that metrizability is an essential assumption in our first claim. This motivates further study to characterize the Bohr group topology as well as detect a monothetic one-parametric topological groups in a non-metrizable setting. Both issues are addressed and resolved in the present paper.