The Baire property in remainders of topological groups and other results

Aleksander V. Arhangel'skii

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 2, page 273-279
  • ISSN: 0010-2628

Abstract

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It is established that a remainder of a non-locally compact topological group G has the Baire property if and only if the space G is not Čech-complete. We also show that if G is a non-locally compact topological group of countable tightness, then either G is submetrizable, or G is the Čech-Stone remainder of an arbitrary remainder Y of G . It follows that if G and H are non-submetrizable topological groups of countable tightness such that some remainders of G and H are homeomorphic, then the spaces G and H are homeomorphic. Some other corollaries and related results are presented.

How to cite

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Arhangel'skii, Aleksander V.. "The Baire property in remainders of topological groups and other results." Commentationes Mathematicae Universitatis Carolinae 50.2 (2009): 273-279. <http://eudml.org/doc/32498>.

@article{Arhangelskii2009,
abstract = {It is established that a remainder of a non-locally compact topological group $G$ has the Baire property if and only if the space $G$ is not Čech-complete. We also show that if $G$ is a non-locally compact topological group of countable tightness, then either $G$ is submetrizable, or $G$ is the Čech-Stone remainder of an arbitrary remainder $Y$ of $G$. It follows that if $G$ and $H$ are non-submetrizable topological groups of countable tightness such that some remainders of $G$ and $H$ are homeomorphic, then the spaces $G$ and $H$ are homeomorphic. Some other corollaries and related results are presented.},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Baire property; $\sigma $-compact; Čech-complete space; compactification; Čech-Stone compactification; Rajkov complete; paracompact $p$-space; topological group; Baire property; compactification; remainder},
language = {eng},
number = {2},
pages = {273-279},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Baire property in remainders of topological groups and other results},
url = {http://eudml.org/doc/32498},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - The Baire property in remainders of topological groups and other results
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 2
SP - 273
EP - 279
AB - It is established that a remainder of a non-locally compact topological group $G$ has the Baire property if and only if the space $G$ is not Čech-complete. We also show that if $G$ is a non-locally compact topological group of countable tightness, then either $G$ is submetrizable, or $G$ is the Čech-Stone remainder of an arbitrary remainder $Y$ of $G$. It follows that if $G$ and $H$ are non-submetrizable topological groups of countable tightness such that some remainders of $G$ and $H$ are homeomorphic, then the spaces $G$ and $H$ are homeomorphic. Some other corollaries and related results are presented.
LA - eng
KW - Baire property; $\sigma $-compact; Čech-complete space; compactification; Čech-Stone compactification; Rajkov complete; paracompact $p$-space; topological group; Baire property; compactification; remainder
UR - http://eudml.org/doc/32498
ER -

References

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  2. Arhangel'skii A.V., 10.1070/RM1981v036n03ABEH004249, Russian Math. Surveys 36 (3) (1981), 151-174. MR0622722DOI10.1070/RM1981v036n03ABEH004249
  3. Arhangel'skii A.V., Moscow spaces and topological groups, Topology Proc. 25 (2000), 383--416. Zbl1027.54038MR1925695
  4. Arhangel'skii A.V., 10.1016/j.topol.2004.10.015, Topology Appl. 150 (2005), 79--90. Zbl1075.54012MR2133669DOI10.1016/j.topol.2004.10.015
  5. Arhangel'skii A.V., Two types of remainders of topological groups, Comment. Math. Univ. Carolin. 49 (2008), no. 1, 119--126. MR2433629
  6. Arhangel'skii A.V., Ponomarev V.I., Fundamentals of General Topology in Problems and Exercises, Reidel, 1984 (translated from Russian). MR0785749
  7. Arhangel'skii A.V., Tkachenko M.G., Topological Groups and Related Structures, Atlantis Press, Paris; World Scientific, Hackensack, NJ, 2008. MR2433295
  8. Choban M.M., On completions of topological groups, Vestnik Moskov. Univ. Ser. Mat. Mech. 1 (1970), 33--38 (in Russian). MR0279226
  9. Choban M.M., Topological structure of subsets of topological groups and their quotients, in Topological Structures and Algebraic Systems, Shtiintsa, Kishinev, 1977, pp. 117--163 (in Russian). 
  10. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  11. Rančin D.V., Tightness, sequentiality, and closed covers, Dokl. Akad. Nauk SSSR 232 (1977), 1015--1018. MR0436074

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