The regularity of quotient paratopological groups

T. Banakh; A. Ravsky

doi:10.15330/ms.49.2.144-149

Let

$H$ be a closed subgroup of a regular Abelian paratopological group

$G$ . The group reflexion

$G^\flat$ of

$G$ is the group

$G$ endowed with the strongest group topology, which is weaker than the original topology of

$G$ . We show that the quotient

$G/H$ is Hausdorff (and regular) if

$H$ is closed (and locally compact) in

$G^\flat$ . On the other hand, we construct an example of a regular abelian paratopological group

$G$ containing a closed discrete subgroup

$H$ such that the quotient

$G/H$ is Hausdorff but not regular.

1. T. Banakh, A. Ravsky, Oscillator topologies on a paratopological group and related number invariants, Algebraical Structures and their Applications, Kyiv: Inst. Mat. NANU, (2002) 140-153.

2. M. Katetov, On H-closed extensions of topological spaces, Casopis Pest. Mat. Fys., 72 (1947), 17-32.

3. A. Ravsky, Paratopological groups I, Mat. Stud., 16 (2001), №1, 37-48.

4. A. Ravsky, Paratopological groups II, Mat. Stud., 17 (2002), №1, 93-101.

5. A. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis, Lviv University, 2002. (in Ukrainian)

6. A. Ravsky, Pseudocompact paratopological groups, http://arxiv.org/abs/1003-5343

7. M. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375-481.

8. M. Sanchis, M. Tkachenko, Totally LindelЃNof and totally !-narrow paratopological groups, Topology Appl., 155 (2008), №4, 322-334.

9. I. Sanchez, M. Tkachenko, Products of bounded subsets of paratopological groups, Topology Appl., 190 (2015), №1, 42-58.

10. Li-Hong Xie, Piyu Li, Jin-Ji Tu, Notes on (regular)

$T_3$ -reflections in the category of semitopological groups, Topology Appl., 178 (2014), 46-55.

	The regularity of quotient paratopological groups
Author	T. Banakh¹, A. Ravsky² tbanakh@yahoo.com¹, oravsky@mail.ru² 1) Department of Mathematics, Ivan Franko Lviv National University Lviv, Ukraine; 2) Department of Functional Analysis Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine Lviv, Ukraine
Abstract	Let $H$ be a closed subgroup of a regular Abelian paratopological group $G$ . The group reflexion $G^\flat$ of $G$ is the group $G$ endowed with the strongest group topology, which is weaker than the original topology of $G$ . We show that the quotient $G/H$ is Hausdorff (and regular) if $H$ is closed (and locally compact) in $G^\flat$ . On the other hand, we construct an example of a regular abelian paratopological group $G$ containing a closed discrete subgroup $H$ such that the quotient $G/H$ is Hausdorff but not regular.
Keywords	paratopological group; quotient paratopological group; group reflexion; regularity
DOI	doi:10.15330/ms.49.2.144-149
Reference	1. T. Banakh, A. Ravsky, Oscillator topologies on a paratopological group and related number invariants, Algebraical Structures and their Applications, Kyiv: Inst. Mat. NANU, (2002) 140-153. 2. M. Katetov, On H-closed extensions of topological spaces, Casopis Pest. Mat. Fys., 72 (1947), 17-32. 3. A. Ravsky, Paratopological groups I, Mat. Stud., 16 (2001), №1, 37-48. 4. A. Ravsky, Paratopological groups II, Mat. Stud., 17 (2002), №1, 93-101. 5. A. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis, Lviv University, 2002. (in Ukrainian) 6. A. Ravsky, Pseudocompact paratopological groups, http://arxiv.org/abs/1003-5343 7. M. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375-481. 8. M. Sanchis, M. Tkachenko, Totally LindelЃNof and totally !-narrow paratopological groups, Topology Appl., 155 (2008), №4, 322-334. 9. I. Sanchez, M. Tkachenko, Products of bounded subsets of paratopological groups, Topology Appl., 190 (2015), №1, 42-58. 10. Li-Hong Xie, Piyu Li, Jin-Ji Tu, Notes on (regular) $T_3$ -reflections in the category of semitopological groups, Topology Appl., 178 (2014), 46-55.
Pages	144-149
Volume	49
Issue	2
Year	2018
Journal	Matematychni Studii
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