Nonnormality of remainders of some topological groups

Aleksander V. Arhangel'skii; J. van Mill

Commentationes Mathematicae Universitatis Carolinae (2016)

  • Volume: 57, Issue: 3, page 345-352
  • ISSN: 0010-2628

Abstract

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It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group G has a normal remainder. In a previous paper we showed that under mild conditions on G , the Continuum Hypothesis implies that if the Čech-Stone remainder G * of G is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight is uncountable but less than 𝔠 , has a normal remainder under 𝖬𝖠 + ¬ 𝖢𝖧 . We also show that if a precompact group with a countable network has a normal remainder, then this group is metrizable. We finally show that if C p ( X ) has a normal remainder, then X is countable (Corollary 4.10) This result provides us with many natural examples of topological groups all remainders of which are nonnormal.

How to cite

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Arhangel'skii, Aleksander V., and Mill, J. van. "Nonnormality of remainders of some topological groups." Commentationes Mathematicae Universitatis Carolinae 57.3 (2016): 345-352. <http://eudml.org/doc/286826>.

@article{Arhangelskii2016,
abstract = {It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group $G$ has a normal remainder. In a previous paper we showed that under mild conditions on $G$, the Continuum Hypothesis implies that if the Čech-Stone remainder $G^*$ of $G$ is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight is uncountable but less than $\mathfrak \{c\}$, has a normal remainder under $\mathsf \{MA\}\{+\}\lnot \mathsf \{CH\}$. We also show that if a precompact group with a countable network has a normal remainder, then this group is metrizable. We finally show that if $C_p(X)$ has a normal remainder, then $X$ is countable (Corollary 4.10) This result provides us with many natural examples of topological groups all remainders of which are nonnormal.},
author = {Arhangel'skii, Aleksander V., Mill, J. van},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {remainder; compactification; topological group; normal space},
language = {eng},
number = {3},
pages = {345-352},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Nonnormality of remainders of some topological groups},
url = {http://eudml.org/doc/286826},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
AU - Mill, J. van
TI - Nonnormality of remainders of some topological groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 3
SP - 345
EP - 352
AB - It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group $G$ has a normal remainder. In a previous paper we showed that under mild conditions on $G$, the Continuum Hypothesis implies that if the Čech-Stone remainder $G^*$ of $G$ is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight is uncountable but less than $\mathfrak {c}$, has a normal remainder under $\mathsf {MA}{+}\lnot \mathsf {CH}$. We also show that if a precompact group with a countable network has a normal remainder, then this group is metrizable. We finally show that if $C_p(X)$ has a normal remainder, then $X$ is countable (Corollary 4.10) This result provides us with many natural examples of topological groups all remainders of which are nonnormal.
LA - eng
KW - remainder; compactification; topological group; normal space
UR - http://eudml.org/doc/286826
ER -

References

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