# 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

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topArhangel'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 -

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