Connected LCA groups are sequentially connected

Shou Lin; Mihail G. Tkachenko

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 2, page 263-272
  • ISSN: 0010-2628

Abstract

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We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if H is a connected locally compact Abelian subgroup of a Hausdorff topological group G and the quotient space G / H is sequentially connected, then so is G .

How to cite

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Lin, Shou, and Tkachenko, Mihail G.. "Connected LCA groups are sequentially connected." Commentationes Mathematicae Universitatis Carolinae 54.2 (2013): 263-272. <http://eudml.org/doc/252467>.

@article{Lin2013,
abstract = {We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if $H$ is a connected locally compact Abelian subgroup of a Hausdorff topological group $G$ and the quotient space $G/H$ is sequentially connected, then so is $G$.},
author = {Lin, Shou, Tkachenko, Mihail G.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {locally compact; connected; sequentially connected; Pontryagin duality; torsion-free; divisible; metrizable element; extension of a group; topological group; locally compact; connected; sequentially connected; Pontryagin duality; torsion-free; divisible; metrizable element; feathered subgroup; quotient space},
language = {eng},
number = {2},
pages = {263-272},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Connected LCA groups are sequentially connected},
url = {http://eudml.org/doc/252467},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Lin, Shou
AU - Tkachenko, Mihail G.
TI - Connected LCA groups are sequentially connected
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 2
SP - 263
EP - 272
AB - We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if $H$ is a connected locally compact Abelian subgroup of a Hausdorff topological group $G$ and the quotient space $G/H$ is sequentially connected, then so is $G$.
LA - eng
KW - locally compact; connected; sequentially connected; Pontryagin duality; torsion-free; divisible; metrizable element; extension of a group; topological group; locally compact; connected; sequentially connected; Pontryagin duality; torsion-free; divisible; metrizable element; feathered subgroup; quotient space
UR - http://eudml.org/doc/252467
ER -

References

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  8. Kuz'minov V., On a hypothesis of P.S. Alexandrov in the theory of topological groups, Dokl. Akad. Nauk SSSR 125 (1959), 727–729 (in Russian). MR0104753
  9. Lin S., The images of connected metric spaces, Chinese Ann. Math. A 26 (2005), 345–350. Zbl1081.54016MR2158870
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