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Each regular paratopological group is completely regular


Authors: Taras Banakh and Alex Ravsky
Journal: Proc. Amer. Math. Soc. 145 (2017), 1373-1382
MSC (2010): Primary 54D10, 54D15, 54E15, 22A30
DOI: https://doi.org/10.1090/proc/13318
Published electronically: September 15, 2016
MathSciNet review: 3589333
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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.


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  • [1] Alexander Arhangelskii, Topological invariants in algebraic environment, Recent progress in general topology, II, North-Holland, Amsterdam, 2002, pp. 1-57. MR 1969992, https://doi.org/10.1016/B978-044450980-2/50001-7
  • [2] Alexander Arhangelskii and Mikhail Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. MR 2433295
  • [3] Taras Banakh and Olexandr Ravsky, Oscillator topologies on a paratopological group and related number invariants, Third International Algebraic Conference in the Ukraine (Ukrainian), Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2002, pp. 140-153. MR 2210489
  • [4] Taras Banakh and Alex Ravsky, On the submetrizability number and $ i$-weight of quasi-uniform spaces and paratopological groups, Topology Proc. 47 (2016), 221-259. MR 3417439
  • [5] Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
  • [6] Peter Fletcher and William F. Lindgren, Quasi-uniform spaces, Lecture Notes in Pure and Applied Mathematics, vol. 77, Marcel Dekker, Inc., New York, 1982. MR 660063
  • [7] Ralph Kopperman, Lengths on semigroups and groups, Semigroup Forum 25 (1982), no. 3-4, 345-360. MR 679288, https://doi.org/10.1007/BF02573609
  • [8] H.-P. Künzi, Quasi-uniform spaces, Encyclopedia of General Topology (eds.: K. P. Hart, J. Nagata, J. Vaughan), Elsevier Sci. Publ., Amsterdam, 2004, pp. 266-270.
  • [9] Hans-Peter A. Künzi, Quasi-uniform spaces in the year 2001, Recent progress in general topology, II, North-Holland, Amsterdam, 2002, pp. 313-344. MR 1970003, https://doi.org/10.1016/B978-044450980-2/50012-1
  • [10] H.-P. A. Künzi, J. Marín, and S. Romaguera, Quasi-uniformities on topological semigroups and bicompletion, Semigroup Forum 62 (2001), no. 3, 403-422. MR 1831463, https://doi.org/10.1007/s002330010033
  • [11] Fucai Lin and Chuan Liu, On paratopological groups, Topology Appl. 159 (2012), no. 10-11, 2764-2773. MR 2923446, https://doi.org/10.1016/j.topol.2012.03.003
  • [12] O. V. Ravsky, Paratopological groups. I, Mat. Stud. 16 (2001), no. 1, 37-48 (English, with English and Russian summaries). MR 1871521
  • [13] O. V. Ravsky, Paratopological groups. II, Mat. Stud. 17 (2002), no. 1, 93-101 (English, with English and Russian summaries). MR 1932275
  • [14] Iván Sánchez, Condensations of paratopological groups, Topology Appl. 180 (2015), 124-131. MR 3293271, https://doi.org/10.1016/j.topol.2014.11.009
  • [15] Walter Roelcke and Susanne Dierolf, Uniform structures on topological groups and their quotients, Advanced Book Program, McGraw-Hill International Book Co., New York, 1981. MR 644485
  • [16] Mikhail Tkachenko, Paratopological and semitopological groups versus topological groups, Recent progress in general topology. III, Atlantis Press, Paris, 2014, pp. 825-882. MR 3205500, https://doi.org/10.2991/978-94-6239-024-9_20
  • [17] M. Tkachenko, Axioms of separation in semitopological groups and related functors, Topology Appl. 161 (2014), 364-376. MR 3132376, https://doi.org/10.1016/j.topol.2013.10.037
  • [18] Li-Hong Xie and Shou Lin, Submetrizability in paratopological groups, Topology Proc. 44 (2014), 139-149. MR 3080645

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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
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
Email: oravsky@mail.ru

DOI: https://doi.org/10.1090/proc/13318
Keywords: Tychonoff space, regular space, completely regular space, semiregular space, Hausdorff space, functionally Hausdorff space, semi-Hausdorff space, separation axiom, quasi-uniformity, paratopological group, topological monoid
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
Article copyright: © Copyright 2016 American Mathematical Society

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