# Group reflection and precompact paratopological groups

Topological Algebra and its Applications (2013)

- Volume: 1, page 22-30
- ISSN: 2299-3231

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topMikhail Tkachenko. "Group reflection and precompact paratopological groups." Topological Algebra and its Applications 1 (2013): 22-30. <http://eudml.org/doc/267177>.

@article{MikhailTkachenko2013,

abstract = {We construct a precompact completely regular paratopological Abelian group G of size (2ω)+ such that all subsets of G of cardinality ≤ 2ω are closed. This shows that Protasov’s theorem on non-closed discrete subsets of precompact topological groups cannot be extended to paratopological groups. We also prove that the group reflection of the product of an arbitrary family of paratopological (even semitopological) groups is topologically isomorphic to the product of the group reflections of the factors, and that the group reflection, H*, of a dense subgroup G of a paratopological group G is topologically isomorphic to a subgroup of G*.},

author = {Mikhail Tkachenko},

journal = {Topological Algebra and its Applications},

keywords = {precompact; pseudocompact; group reflection; paratopological group},

language = {eng},

pages = {22-30},

title = {Group reflection and precompact paratopological groups},

url = {http://eudml.org/doc/267177},

volume = {1},

year = {2013},

}

TY - JOUR

AU - Mikhail Tkachenko

TI - Group reflection and precompact paratopological groups

JO - Topological Algebra and its Applications

PY - 2013

VL - 1

SP - 22

EP - 30

AB - We construct a precompact completely regular paratopological Abelian group G of size (2ω)+ such that all subsets of G of cardinality ≤ 2ω are closed. This shows that Protasov’s theorem on non-closed discrete subsets of precompact topological groups cannot be extended to paratopological groups. We also prove that the group reflection of the product of an arbitrary family of paratopological (even semitopological) groups is topologically isomorphic to the product of the group reflections of the factors, and that the group reflection, H*, of a dense subgroup G of a paratopological group G is topologically isomorphic to a subgroup of G*.

LA - eng

KW - precompact; pseudocompact; group reflection; paratopological group

UR - http://eudml.org/doc/267177

ER -

## References

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- [2] T. Banakh and O. Ravsky, Oscillator topologies on a paratopological group and related number invariants, Algebraic Structures and their Applications, Kyiv: Inst. Mat. NANU, (2002), 140-152. Zbl1098.22004
- [3] W.W. Comfort, and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483–496. Zbl0214.28502
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- [5] R. Engelking, General Topology, Heldermann Verlag, Berlin 1989.
- [6] M. Fernández, On some classes of paratopological groups, Topology Proc. 40 (2012), 63–72. Zbl1271.54068
- [7] L. S. Pontryagin, Continuous groups, third edition, “Nauka”, Moscow 1973.
- [8] I. V. Protasov, Discrete subsets of topological groups, Math. Notes 55 (1994) no. 1–2, 101–102. Russian original in: Mat. Zametki 55 (1994), 150–151. Zbl0836.22003
- [9] O. V. Ravsky, Paratopological groups, II, Mat. Studii 17 (2002), no. 1, 93–101. [WoS] Zbl1018.22001
- [10] M. G. Tkachenko, Paratopological Groups: Some Questions and Problems, Q&A in General Topology 27 no. 1 (2009), 1–21. Zbl1173.54315

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