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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

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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

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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

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  12. [12] O.V. Ravsky, Pseudocompact paratopological groups that are topological, http://arxiv.org/abs/1003.5343 (April 7, 2012). 
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