Each regular paratopological group is completely regular
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- by Taras Banakh and Alex Ravsky PDF
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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.References
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Additional Information
- Taras Banakh
- Affiliation: Department of Mathematics, Ivan Franko National University of Lviv, Lviv, Ukraine 79000 – and – Jan Kochanowski University in Kielce, Poland
- MR Author ID: 249694
- Email: t.o.banakh@gmail.com
- Alex Ravsky
- Affiliation: Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences, Lviv, Ukraine 79060
- MR Author ID: 647500
- Email: oravsky@mail.ru
- Received by editor(s): March 17, 2015
- Received by editor(s) in revised form: May 15, 2016
- Published electronically: September 15, 2016
- Additional Notes: The first author has been partially financed by NCN grant DEC-2012/07/D/ST1/02087.
- Communicated by: Mirna Džamonja
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1373-1382
- MSC (2010): Primary 54D10, 54D15, 54E15, 22A30
- DOI: https://doi.org/10.1090/proc/13318
- MathSciNet review: 3589333