| Title:
|
A note on topological groups and their remainders (English) |
| Author:
|
Peng, Liang-Xue |
| Author:
|
He, Yu-Feng |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
197-214 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
topological group |
| Keyword:
|
remainder |
| Keyword:
|
compactification |
| Keyword:
|
metrizable space |
| Keyword:
|
weak base |
| MSC:
|
54A25 |
| MSC:
|
54B05 |
| MSC:
|
54D40 |
| MSC:
|
54E99 |
| idZBL:
|
Zbl 1249.54058 |
| idMR:
|
MR2899745 |
| DOI:
|
10.1007/s10587-012-0005-x |
| . |
| Date available:
|
2012-03-05T07:24:58Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142051 |
| . |
| Reference:
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