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 |

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Category: | math |

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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 |

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Date available: | 2009-01-08T18:57:06Z |

Last updated: | 2012-04-30 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/119127 |

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