| Title:
|
Connectedness and local connectedness of topological groups and extensions (English) |
| Author:
|
Alas, O. T. |
| Author:
|
Tkačenko, M. G. |
| Author:
|
Tkachuk, V. V. |
| Author:
|
Wilson, R. G. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
4 |
| Year:
|
1999 |
| Pages:
|
735-753 |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is shown that both the free topological group $F(X)$ and the free Abelian topological group $A(X)$ on a connected locally connected space $X$ are locally connected. For the Graev's modification of the groups $F(X)$ and $A(X)$, the corresponding result is more symmetric: the groups $F\Gamma(X)$ and $A\Gamma(X)$ are connected and locally connected if $X$ is. However, the free (Abelian) totally bounded group $FTB(X)$ (resp., $ATB(X)$) is not locally connected no matter how ``good'' a space $X$ is. The above results imply that every non-trivial continuous homomorphism of $A(X)$ to the additive group of reals, with $X$ connected and locally connected, is open. We also prove that any dense in itself subspace of the Sorgenfrey line has a Urysohn connectification. If $D$ is a dense subset of $\{0,1\}^{\frak c}$ of power less than $\frak c$, then $D$ has a Urysohn connectification of the same cardinality as $D$. We also strengthen a result of [1] for second countable Tychonoff spaces without open compact subspaces proving that it is possible to find a compact metrizable connectification of such a space preserving its dimension if it is positive. (English) |
| Keyword:
|
connected |
| Keyword:
|
locally connected |
| Keyword:
|
free topological group |
| Keyword:
|
Pontryagin's duality |
| Keyword:
|
pseudo-open mapping |
| Keyword:
|
open mapping |
| Keyword:
|
Urysohn space |
| Keyword:
|
connectification |
| MSC:
|
22A05 |
| MSC:
|
54C10 |
| MSC:
|
54C25 |
| MSC:
|
54D06 |
| MSC:
|
54D25 |
| MSC:
|
54H11 |
| idZBL:
|
Zbl 1010.54043 |
| idMR:
|
MR1756549 |
| . |
| Date available:
|
2009-01-08T18:57:06Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119127 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| . |
