# A note on topological groups and their remainders

Czechoslovak Mathematical Journal (2012)

- Volume: 62, Issue: 1, page 197-214
- ISSN: 0011-4642

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topPeng, Liang-Xue, and He, Yu-Feng. "A note on topological groups and their remainders." Czechoslovak Mathematical Journal 62.1 (2012): 197-214. <http://eudml.org/doc/247202>.

@article{Peng2012,

abstract = {In this note we first give a summary that on property of a remainder of a non-locally compact topological group $G$ in a compactification $bG$ makes the remainder and the topological group $G$ all separable and metrizable. If a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ of $G$ belongs to $\mathcal \{P\}$, then $G$ and $bG\setminus G$ are separable and metrizable, where $\mathcal \{P\}$ is a class of spaces which satisfies the following conditions: (1) if $X\in \mathcal \{P\}$, then every compact subset of the space $X$ is a $G_\delta $-set of $X$; (2) if $X\in \mathcal \{P\}$ and $X$ is not locally compact, then $X$ is not locally countably compact; (3) if $X\in \mathcal \{P\}$ and $X$ is a Lindelöf $p$-space, then $X$ is metrizable. Some known conclusions on topological groups and their remainders can be obtained from this conclusion. As a corollary, we have that if a non-locally compact topological group $G$ has a compactification $bG$ such that compact subsets of $bG\setminus G$ are $G_\{\delta \}$-sets in a uniform way (i.e., $bG\setminus G$ is CSS), then $G$ and $bG\setminus G$ are separable and metrizable spaces. In the last part of this note, we prove that if a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ has a point-countable weak base and has a dense subset $D$ such that every point of the set $D$ has countable pseudo-character in the remainder $bG\setminus G$ (or the subspace $D$ has countable $\pi $-character), then $G$ and $bG\setminus G$ are both separable and metrizable.},

author = {Peng, Liang-Xue, He, Yu-Feng},

journal = {Czechoslovak Mathematical Journal},

keywords = {topological group; remainder; compactification; metrizable space; weak base; topological group; compactification; remainder; metrizability; cardinal function; weak base},

language = {eng},

number = {1},

pages = {197-214},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {A note on topological groups and their remainders},

url = {http://eudml.org/doc/247202},

volume = {62},

year = {2012},

}

TY - JOUR

AU - Peng, Liang-Xue

AU - He, Yu-Feng

TI - A note on topological groups and their remainders

JO - Czechoslovak Mathematical Journal

PY - 2012

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 62

IS - 1

SP - 197

EP - 214

AB - In this note we first give a summary that on property of a remainder of a non-locally compact topological group $G$ in a compactification $bG$ makes the remainder and the topological group $G$ all separable and metrizable. If a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ of $G$ belongs to $\mathcal {P}$, then $G$ and $bG\setminus G$ are separable and metrizable, where $\mathcal {P}$ is a class of spaces which satisfies the following conditions: (1) if $X\in \mathcal {P}$, then every compact subset of the space $X$ is a $G_\delta $-set of $X$; (2) if $X\in \mathcal {P}$ and $X$ is not locally compact, then $X$ is not locally countably compact; (3) if $X\in \mathcal {P}$ and $X$ is a Lindelöf $p$-space, then $X$ is metrizable. Some known conclusions on topological groups and their remainders can be obtained from this conclusion. As a corollary, we have that if a non-locally compact topological group $G$ has a compactification $bG$ such that compact subsets of $bG\setminus G$ are $G_{\delta }$-sets in a uniform way (i.e., $bG\setminus G$ is CSS), then $G$ and $bG\setminus G$ are separable and metrizable spaces. In the last part of this note, we prove that if a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ has a point-countable weak base and has a dense subset $D$ such that every point of the set $D$ has countable pseudo-character in the remainder $bG\setminus G$ (or the subspace $D$ has countable $\pi $-character), then $G$ and $bG\setminus G$ are both separable and metrizable.

LA - eng

KW - topological group; remainder; compactification; metrizable space; weak base; topological group; compactification; remainder; metrizability; cardinal function; weak base

UR - http://eudml.org/doc/247202

ER -

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