Group reflection and precompact paratopological groups

Mikhail Tkachenko

Topological Algebra and its Applications (2013)

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

Abstract

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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*.

How to cite

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Mikhail 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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  1. [1] A. V. Arhangel’skii and M. G. Tkachenko, Topological Groups and Related Structures, Atlantis Series in Mathematics, vol. I, Atlantis Press and World Scientific, Paris–Amsterdam 2008. 
  2. [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. [3] W.W. Comfort, and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483–496. Zbl0214.28502
  4. [4] S. Dierolf and U. Schwanengel, Examples of locally compact non-compact minimal topological groups, Pacific J. Math. 82 (1979), 349–355. Zbl0388.22002
  5. [5] R. Engelking, General Topology, Heldermann Verlag, Berlin 1989. 
  6. [6] M. Fernández, On some classes of paratopological groups, Topology Proc. 40 (2012), 63–72. Zbl1271.54068
  7. [7] L. S. Pontryagin, Continuous groups, third edition, “Nauka”, Moscow 1973. 
  8. [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. [9] O. V. Ravsky, Paratopological groups, II, Mat. Studii 17 (2002), no. 1, 93–101. [WoS] Zbl1018.22001
  10. [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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