Cardinal invariants of paratopological groups

• Autorzy: Iván Sánchez
• Języki publikacji: EN

Abstrakty

EN

We show that a regular totally ω-narrow paratopological group G has countable index of regularity, i.e., for every neighborhood U of the identity e of G, we can find a neighborhood V of e and a countable family of neighborhoods of e in G such that ∩W∈γ VW−1⊆ U. We prove that every regular (Hausdorff) totally !-narrow paratopological group is completely regular (functionally Hausdorff). We show that the index of regularity of a regular paratopological group is less than or equal to the weak Lindelöf number. We also prove that every Hausdorff paratopological group with countable π- character has a regular Gσ-diagonal.

Słowa kluczowe

EN

Paratopological group; Totally ω-narrow; Index of regularity; Weak Lindelöf number; Hausdorff number; Symmetry number; Regular Gσ-diagonal

Kategorie tematyczne

• 54B10: Product spaces
• 54D10: Lower separation axioms ( T 0 -- T 3 , etc.)
• 54A25: Cardinality properties (cardinal functions and inequalities, discrete subsets)
• 54H11: Topological groups

• Wydawca: De Gruyter Open
• Czasopismo: Topological Algebra and its Applications
• Rocznik: 2013
• Tom: 1
• Strony: 37-45

Daty

otrzymano

2013-04-10

zaakceptowano

2013-05-02

online

2013-12-21

Twórcy

autor

Iván Sánchez

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

Bibliografia

• [1] A. V. Arhangel’skii, D. K. Burke, Spaces with a regular Gσ-diagonal, Topol. Appl. 153 No. 11 (2006), 1917–1929.
• [2] A.V. Arhangel’skii, E.A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology Appl. 151 (2005) 107–119. [WoS]
• [3] A.V. Arhangel’skii, M.G. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, Vol. I, Atlantis Press/World Scientific, Paris-Amsterdam, 2008.
• [4] T. Banakh, A. Ravsky, On subgroups of saturated or totally bounded paratopological groups, Algebra and Discrete Mathematics No. 4 (2003) 1–20.
• [5] T. Banakh, O. Ravsky, Oscillator topologies on a paratopological group and related number invariants, Algebraical Structures and their Applications, Kyiv: Inst. Mat. NANU (2002), 140–153.
• [6] R. Z. Buzyakova, Observations on spaces with zeroset or regular Gσ-diagonals, Comment. Math. Univ. Carolin. 46 (2005), No. 3, 469–473.
• [7] R. Engelkig, General Topology, Heldermann Verlag, Berlin, 1989.
• [8] C. Hernández, Condensations of Tychonoff universal topological algebras, Comment. Math. Univ. Carolin. 42 (2001), No. 3, 529–533.
• [9] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I, Structure of Topological Groups, Integration Theory, Group Representations. Second edition. Fund. Prin. of Math. Sci., 115. (Springer-Verlag, Berlin-New York, 1979).
• [10] I. Juhász, Cardinal functions in topology–ten years later, second ed., Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, Amsterdam, 1980.
• [11] P. Li, L. Mou, S. Wang, Notes on questions about spaces with algebraic structures, Topology Appl. 159 (2012), 3619–3623. [WoS]
• [12] C. Liu, A note on paratopological groups, Comment. Math. Univ. Carolin. 47 No. 4 (2006), 633–640.
• [13] O.V. Ravsky, Paratopological groups I, Matematychni Studii 16 (2001), No. 1, 37–48.
• [14] O.V. Ravsky, Paratopological groups II, Matematychni Studii 17 (2002), No. 1, 93–101.
• [15] D.B. Shakhmatov, Condensation of universal topological algebras that preserve continuity of operations and decrease weight, Vestnik Mosk. Univ. 39 (1984), 42–45.
• [16] I. Sánchez, Subgroups of products of paratopological groups, Topology Apply., to appear. [WoS]
• [17] M. Sanchis, M.G. Tkachenko, Recent progress in paratopological groups, Quaderni Math. (2012), in press.
• [18] M. Sanchis, M.G. Tkachenko, Totally Lindelöf and totally !-narrow paratopological groups, Topology Apply. 155 (2007) 322–334.
• [19] M.G. Tkachenko, Introduction to topological groups, Topology Appl. 86 (1998) 179–231. [Crossref]
• [20] M.G. Tkachenko, Embedding paratopological groups into topological products, Topology Appl. 156 (2009) 1298– 1305. [WoS]
• [21] M.G Tkachenko, Paratopological and semitopological groups vs topological groups, In: Recent Progress in General Topology III, Elsevier, to appear.
• [22] M.G. Tkachenko, Paratopological Groups: Some Questions and Problems, Q&A in General Topology 27 No. 1 (2009), 1–21.
• [23] L.H. Xie, S. Lin, Submetrizability in paratopological groups, submitted.
• [24] P. Zenor, On spaces with regular Gσ-diagonals, Pacific J. Math. 40 (1972), 759–763.

Identyfikatory

DOI

10.2478/taa-2013-0005