A note on pseudobounded paratopological groups

Fucai Lin; Shou Lin; Iván Sánchez

Topological Algebra and its Applications (2014)

  • Volume: 2, Issue: 1, page 11-18, electronic only
  • ISSN: 2299-3231

Abstract

top
Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudobounded paratopological group such that G is a Lusin space, then is G a topological group. We present some examples of paratopological groups with interesting properties: (1) There exists a metrizable, zero-dimensional and pseudobounded topological group; (2) There exists a Hausdorff ω-pseudobounded paratopological group G such that G contains a dense subgroup which is not ω-pseudobounded; (3) There exists a Hausdorff connected paratopological group which is not ω-pseudobounded.

How to cite

top

Fucai Lin, Shou Lin, and Iván Sánchez. "A note on pseudobounded paratopological groups." Topological Algebra and its Applications 2.1 (2014): 11-18, electronic only. <http://eudml.org/doc/267456>.

@article{FucaiLin2014,
abstract = {Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudobounded paratopological group such that G is a Lusin space, then is G a topological group. We present some examples of paratopological groups with interesting properties: (1) There exists a metrizable, zero-dimensional and pseudobounded topological group; (2) There exists a Hausdorff ω-pseudobounded paratopological group G such that G contains a dense subgroup which is not ω-pseudobounded; (3) There exists a Hausdorff connected paratopological group which is not ω-pseudobounded.},
author = {Fucai Lin, Shou Lin, Iván Sánchez},
journal = {Topological Algebra and its Applications},
keywords = {Paratopological group; Pseudobounded; ω-pseudobounded; Topological group; Premeager space; Lusin space; paratopological group; pseudobounded; -pseudobounded; topological group; premeager space},
language = {eng},
number = {1},
pages = {11-18, electronic only},
title = {A note on pseudobounded paratopological groups},
url = {http://eudml.org/doc/267456},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Fucai Lin
AU - Shou Lin
AU - Iván Sánchez
TI - A note on pseudobounded paratopological groups
JO - Topological Algebra and its Applications
PY - 2014
VL - 2
IS - 1
SP - 11
EP - 18, electronic only
AB - Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudobounded paratopological group such that G is a Lusin space, then is G a topological group. We present some examples of paratopological groups with interesting properties: (1) There exists a metrizable, zero-dimensional and pseudobounded topological group; (2) There exists a Hausdorff ω-pseudobounded paratopological group G such that G contains a dense subgroup which is not ω-pseudobounded; (3) There exists a Hausdorff connected paratopological group which is not ω-pseudobounded.
LA - eng
KW - Paratopological group; Pseudobounded; ω-pseudobounded; Topological group; Premeager space; Lusin space; paratopological group; pseudobounded; -pseudobounded; topological group; premeager space
UR - http://eudml.org/doc/267456
ER -

References

top
  1. [1] O.T. Alas, M. Sanchis, Countably compact paratopological groups, Semigroup Forum 74 (2007) 423–438. [WoS] Zbl1125.22001
  2. [2] A.V. Arhangel’skii, E.A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology Appl. 151 (2005) 107–119. Zbl1077.54023
  3. [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. [4] K. H. Azar, Bounded topological groups, arXiv: 1003.2876. 
  5. [5] T. Banakh, I. Guran, O. Ravsky, Characterizing meager paratopological groups, Applied General Topology, 12(1)(2011) 27– 33. Zbl1216.22003
  6. [6] R. Engelkig, General Topology, Heldermann Verlag, Berlin, 1989. 
  7. [7] 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). Zbl0416.43001
  8. [8] F. Lin and S. Lin, Pseudobounded or !-pseudobounded paratopological groups, Filomat, 25:3 (2011) 93–103. [WoS] Zbl1265.54133
  9. [9] F. Lin and C. Liu, On paratopological groups, Topology Apply., 159 (2012) 2764–2773. [WoS] Zbl1276.54027
  10. [10] O.V. Ravsky, Paratopological groups I, Matematychni Studii 16 (2001), No. 1, 37–48. [WoS] Zbl0989.22007
  11. [11] O.V. Ravsky, Paratopological groups II, Matematychni Studii 17 (2002), No. 1, 93–101. [WoS] Zbl1018.22001
  12. [12] O.V. Ravsky, Pseudocompact paratopological groups that are topological, http://arxiv.org/abs/1003.5343 (April 7, 2012). 
  13. [13] M. Sanchis, M.G. Tkachenko, Totally Lindelöf and totally !-narrow paratopological groups, Topology Apply. 155 (2007) 322–334. 
  14. [14] M. Tkachenko, Group reflection and precompact paratopological groups, Topological Algebra and its Applications, (2013) 22–30. 
  15. [15] M. Tkachenko, Embedding paratopological groups into topological products, Topol. Appl., 156(2009) 1298-1305. [WoS] Zbl1166.54016

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.