A note on compact-like semitopological groups
We present a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is provided a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topologi...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2019-12-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
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| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/2122 |
| _version_ | 1818408893230874624 |
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| author | A. Ravsky |
| author_facet | A. Ravsky |
| author_sort | A. Ravsky |
| collection | DOAJ |
| description | We present a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is provided a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular $T_1$ compact and $T_2$ sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group $(G,\tau)$ is a topological group provided there exists a Hausdorff topology $\sigma\supset\tau$ on $G$ such that $(G,\sigma)$ is a precompact topological group and $(G,\tau)$ is weakly semiregular or $(G,\sigma)$ is a feebly compact paratopological group and $(G,\tau)$ is $T_3$. |
| first_indexed | 2024-12-14T09:50:58Z |
| format | Article |
| id | doaj.art-3373d1f2e9e4488cad41c4f6d8190154 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-12-14T09:50:58Z |
| publishDate | 2019-12-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-3373d1f2e9e4488cad41c4f6d81901542022-12-21T23:07:31ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102019-12-0111244245210.15330/cmp.11.2.442-4522122A note on compact-like semitopological groupsA. Ravsky0Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, UkraineWe present a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is provided a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular $T_1$ compact and $T_2$ sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group $(G,\tau)$ is a topological group provided there exists a Hausdorff topology $\sigma\supset\tau$ on $G$ such that $(G,\sigma)$ is a precompact topological group and $(G,\tau)$ is weakly semiregular or $(G,\sigma)$ is a feebly compact paratopological group and $(G,\tau)$ is $T_3$.https://journals.pnu.edu.ua/index.php/cmp/article/view/2122semitopological groupparatopological groupcompact-like semitopological groupcompact-like paratopological groupcontinuity of the inversejoint continuityseparation axiomscountably compact paratopological groupfeebly compact topological group |
| spellingShingle | A. Ravsky A note on compact-like semitopological groups Karpatsʹkì Matematičnì Publìkacìï semitopological group paratopological group compact-like semitopological group compact-like paratopological group continuity of the inverse joint continuity separation axioms countably compact paratopological group feebly compact topological group |
| title | A note on compact-like semitopological groups |
| title_full | A note on compact-like semitopological groups |
| title_fullStr | A note on compact-like semitopological groups |
| title_full_unstemmed | A note on compact-like semitopological groups |
| title_short | A note on compact-like semitopological groups |
| title_sort | note on compact like semitopological groups |
| topic | semitopological group paratopological group compact-like semitopological group compact-like paratopological group continuity of the inverse joint continuity separation axioms countably compact paratopological group feebly compact topological group |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/2122 |
| work_keys_str_mv | AT aravsky anoteoncompactlikesemitopologicalgroups AT aravsky noteoncompactlikesemitopologicalgroups |