Connectedness and local connectedness of topological groups and extensions
Alas, Ofelia Teresa ; Tkachenko, Mihail G. ; Tkachuk, Vladimir Vladimirovich ; Wilson, Richard Gordon
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999), p. 735-753 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 1999-01-01
Classification:  22A05,  54C10,  54C25,  54D06,  54D25,  54H11 
@article{119127,
     author = {Ofelia Teresa Alas and Mihail G. Tkachenko and Vladimir Vladimirovich Tkachuk and Richard Gordon Wilson},
     title = {Connectedness and local connectedness of topological groups and extensions},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {40},
     year = {1999},
     pages = {735-753},
     zbl = {1010.54043},
     mrnumber = {1756549},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119127}
}

Alas, Ofelia Teresa; Tkachenko, Mihail G.; Tkachuk, Vladimir Vladimirovich; Wilson, Richard Gordon. Connectedness and local connectedness of topological groups and extensions. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 735-753. http://gdmltest.u-ga.fr/item/119127/

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