On pseudobounded and premeage paratopological groups

  • A.V. Ravsky Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine
  • T.O. Banakh Ivan Franko National University of Lviv, Lviv, Ukraine
Keywords: topologized group; paratopological group

Abstract

Let $G$ be a paratopological group.
Following F.~Lin and S.~Lin, we say that the group $G$ is pseudobounded,
if for any neighborhood $U$ of the identity of $G$,
there exists a natural number $n$ such that $U^n=G$.
The group $G$ is $\omega$-pseudobounded,
if for any neighborhood $U$ of the identity of $G$, the group $G$ is a
union of sets $U^n$, where $n$ is a natural number.
The group $G$ is premeager, if $G\ne N^n$ for any nowhere dense subset $N$ of
$G$ and any positive integer $n$.
In this paper we investigate relations between the above classes of groups and
answer some questions posed by F. Lin, S. Lin, and S\'anchez.

References

1. A. Arhangel’skiı̆, M. Tkachenko, Topological groups and related sructures, Atlantis Press, 2008.
2. K.H. Azar, Bounded topological groups, arxiv.org/abs/1003.2876
3. T. Banakh, S. Gla̧b, E. Jablońska, J. Swaczyna, Haar-I sets: looking at small sets in Polish groups through compact glasses, Dissert. Math., 564 (2021), 1–105. arxiv.org/abs/1803.06712v4
4. T. Banakh, A. Ravsky, On feebly compact paratopological groups, Topology Appl., 284 (2020), 107363. www.sciencedirect.com/science/article/abs/pii/S0166864120303060
5. W.W. Comfort, S.A. Morris, D. Robbie, S. Svetlichny, M. Tkachenko, Suitable sets for topological groups, Topol. Appl., 86 (1998), 25–46.
6. D. Dikranjan, I. Prodanov, L. Stoyanov, Topological groups: characters dualities and minimal group topologies, (2nd edn.), Monographs and Textbooks in Pure and Applied Mathematics, V.130, Marcel Dekker, New York, 1989.
7. D. Dikranjan, M. Tkachenko, V. Tkachuk, Some topological groups with and some without suitable sets, Topol. Appl., 98 (1999), 131–148.
8. D. Dikranjan, M. Tkačenko, V. Tkachuk, Topological groups with thin generating sets, J. Pure Appl. Algebra, 145 (2000), No2 123–148.
9. I. Guran, Suitable sets for paratopological groups, Abstracts of 4-th International Algebraic Conference in Ukraine (Lviv, 2003), 87–88. prima.lnu.edu.ua/faculty/mechmat/Departments/Topology/Gutik.files/Fourth%20International%20Algebraic%20Conference%20in%20Ukraine.pdf
10. K.H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra, 68 (1990), 181–194.
11. A.S. Kechris, Classical descriptive set theory, Springer, 1995.
12. K. Kunen, Luzin spaces, Topology Proc. I (Conf., Auburn Univ., Auburn, Ala., 1976), (1977) 191–199. www.topology.auburn.edu/tp/reprints/v01/tp01021.pdf
13. F. Lin, Sh. Lin, Pseudobounded or ω-pseudobounded paratopological groups, Filomat, 25 (2011), No3, 93–103. www.doiserbia.nb.rs/img/doi/0354-5180/2011/0354-51801103093L.pdf
14. F. Lin, Sh. Lin, I. Sánchez, A note on pseudobounded paratopological groups, Topol. Algebra Appl., 2 (2014), 11–18. www.ndsy.cn/jsfc/linsuo/LSPapers/LSp192.pdf
15. F. Lin, A. Ravsky, T. Shi, Suitable sets for paratopological groups, 115 (2021), 183. doi.org/10.1007/s13398-021-01129-w
16. N.N. Lusin, Sur un problème de M. Baire, C.R. Acad. Sci. Paris, 158 (1914), 1258–1261.
17. L. Pontrjagin, Continuous groups, Nauka, Moscow, 1973, in Russian.
18. I. Protasov, T. Banakh, Ball structures and colorings of graphs and groups, VNTL Publ., Lviv, 2003.
19. A. Ravsky, Paratopological groups I, Mat. Stud., 16 (2001), No1, 37–48. http://matstud.org.ua/texts/2001/16_1/37_48.pdf
20. A. Ravsky, Paratopological groups II, Mat. Stud., 17 (2002), No1, 93–101. http://matstud.org.ua/texts/2002/17_1/93_101.pdf
21. A. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis, Lviv University, Lviv, 2002, in Ukrainian.
22. M. Tkachenko, Generating dense subgroups of topological groups, Topology Proc., 22 (1997), 533–582
23. M. Tkachenko, Semitopological and paratopological groups vs topological groups, In: Recent Progress in General Topology III (K.P. Hart, J. van Mill, P. Simon, eds.), 2013, 803–859.
24. A. Weil, Sur les Espaces à Structure Uniforme et sur la Topologie Générale, Publications in Mathematics University of Strasbourg, V.551, Hermann & Cie, Paris (1938).
Published
2021-10-27
How to Cite
1.
Ravsky A, Banakh T. On pseudobounded and premeage paratopological groups. Mat. Stud. [Internet]. 2021Oct.27 [cited 2022Jan.20];56(1):20-7. Available from: http://www.matstud.org.ua/ojs/index.php/matstud/article/view/261
Section
Articles