Abstract
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.
References
[1] A. V. Arhangel’skii, D. K. Burke, Spaces with a regular Gσ-diagonal, Topol. Appl. 153 No. 11 (2006), 1917–1929. 10.1016/j.topol.2005.07.013Search in Google Scholar
[2] A.V. Arhangel’skii, E.A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology Appl. 151 (2005) 107–119. 10.1016/j.topol.2003.08.035Search in Google Scholar
[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. 10.2991/978-94-91216-35-0Search in Google Scholar
[4] T. Banakh, A. Ravsky, On subgroups of saturated or totally bounded paratopological groups, Algebra and Discrete Mathematics No. 4 (2003) 1–20. Search in Google Scholar
[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. Search in Google Scholar
[6] R. Z. Buzyakova, Observations on spaces with zeroset or regular Gσ-diagonals, Comment. Math. Univ. Carolin. 46 (2005), No. 3, 469–473. Search in Google Scholar
[7] R. Engelkig, General Topology, Heldermann Verlag, Berlin, 1989. Search in Google Scholar
[8] C. Hernández, Condensations of Tychonoff universal topological algebras, Comment. Math. Univ. Carolin. 42 (2001), No. 3, 529–533. Search in Google Scholar
[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). Search in Google Scholar
[10] I. Juhász, Cardinal functions in topology–ten years later, second ed., Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, Amsterdam, 1980. Search in Google Scholar
[11] P. Li, L. Mou, S. Wang, Notes on questions about spaces with algebraic structures, Topology Appl. 159 (2012), 3619–3623. 10.1016/j.topol.2012.09.001Search in Google Scholar
[12] C. Liu, A note on paratopological groups, Comment. Math. Univ. Carolin. 47 No. 4 (2006), 633–640. Search in Google Scholar
[13] O.V. Ravsky, Paratopological groups I, Matematychni Studii 16 (2001), No. 1, 37–48. Search in Google Scholar
[14] O.V. Ravsky, Paratopological groups II, Matematychni Studii 17 (2002), No. 1, 93–101. Search in Google Scholar
[15] D.B. Shakhmatov, Condensation of universal topological algebras that preserve continuity of operations and decrease weight, Vestnik Mosk. Univ. 39 (1984), 42–45. Search in Google Scholar
[16] I. Sánchez, Subgroups of products of paratopological groups, Topology Apply., to appear. Search in Google Scholar
[17] M. Sanchis, M.G. Tkachenko, Recent progress in paratopological groups, Quaderni Math. (2012), in press. Search in Google Scholar
[18] M. Sanchis, M.G. Tkachenko, Totally Lindelöf and totally !-narrow paratopological groups, Topology Apply. 155 (2007) 322–334. 10.1016/j.topol.2007.05.017Search in Google Scholar
[19] M.G. Tkachenko, Introduction to topological groups, Topology Appl. 86 (1998) 179–231. 10.1016/S0166-8641(98)00051-0Search in Google Scholar
[20] M.G. Tkachenko, Embedding paratopological groups into topological products, Topology Appl. 156 (2009) 1298– 1305. 10.1016/j.topol.2008.12.032Search in Google Scholar
[21] M.G Tkachenko, Paratopological and semitopological groups vs topological groups, In: Recent Progress in General Topology III, Elsevier, to appear. Search in Google Scholar
[22] M.G. Tkachenko, Paratopological Groups: Some Questions and Problems, Q&A in General Topology 27 No. 1 (2009), 1–21. Search in Google Scholar
[23] L.H. Xie, S. Lin, Submetrizability in paratopological groups, submitted. Search in Google Scholar
[24] P. Zenor, On spaces with regular Gσ-diagonals, Pacific J. Math. 40 (1972), 759–763. 10.2140/pjm.1972.40.759Search in Google Scholar
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