A note on topological groups and their remainders

Peng, 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 -