Each regular paratopological group is completely regular

Banakh, Taras; Ravsky, Alex

doi:10.1090/proc/13318

Each regular paratopological group is completely regular
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by Taras Banakh and Alex Ravsky PDF

Proc. Amer. Math. Soc. 145 (2017), 1373-1382 Request permission

Abstract:

We prove that a semiregular topological space $X$ 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.

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