Group reflection and precompact paratopological groups

• Autorzy: Mikhail Tkachenko
• Języki publikacji: EN

Abstrakty

EN

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*.

Słowa kluczowe

EN

precompact; pseudocompact; group reflection; paratopological group

Kategorie tematyczne

• 54D30: Compactness
• 43A40: Character groups and dual objects
• 54G20: Counterexamples
• 22A05: Structure of general topological groups
• 54H11: Topological groups

• Wydawca: De Gruyter Open
• Czasopismo: Topological Algebra and its Applications
• Rocznik: 2013
• Tom: 1
• Strony: 22-30

Daty

otrzymano

2012-12-10

zaakceptowano

2013-03-20

online

2013-07-16

Twórcy

autor

Mikhail Tkachenko

• Departamento de Matemáticas, Universidad Autónoma Metropolitana
  Av. San Rafael Atlixco 186, Col. Vicentina, Iztapalapa C.P. 09340,
  México, D.F., Mexico

Bibliografia

• [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] 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.
• [3] W.W. Comfort, and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483–496.
• [4] S. Dierolf and U. Schwanengel, Examples of locally compact non-compact minimal topological groups, Pacific J. Math. 82 (1979), 349–355.
• [5] R. Engelking, General Topology, Heldermann Verlag, Berlin 1989.
• [6] M. Fernández, On some classes of paratopological groups, Topology Proc. 40 (2012), 63–72.
• [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.
• [9] O. V. Ravsky, Paratopological groups, II, Mat. Studii 17 (2002), no. 1, 93–101. [WoS]
• [10] M. G. Tkachenko, Paratopological Groups: Some Questions and Problems, Q&A in General Topology 27 no. 1 (2009), 1–21.

Identyfikatory

DOI

10.2478/taa-2013-0003