Each regular paratopological group is completely regular
General Topology
2021-11-01 v9 Group Theory
Abstract
We prove that a semiregular topological space is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.
Cite
@article{arxiv.1410.1504,
title = {Each regular paratopological group is completely regular},
author = {Taras Banakh and Alex Ravsky},
journal= {arXiv preprint arXiv:1410.1504},
year = {2021}
}
Comments
7 pages (in the updated version some results on the submetrizability of paratopological groups are added)