Connectedness and local connectedness of topological groups and extensions

Alas, Ofelia Teresa, et al. "Connectedness and local connectedness of topological groups and extensions." Commentationes Mathematicae Universitatis Carolinae 40.4 (1999): 735-753. <http://eudml.org/doc/248437>.

@article{Alas1999,
abstract = {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 $\lbrace 0,1\rbrace ^\{\mathfrak \{c\}\}$ of power less than $\mathfrak \{c\}$, 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.},
author = {Alas, Ofelia Teresa, Tkachenko, Mihail G., Tkachuk, Vladimir Vladimirovich, Wilson, Richard Gordon},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {connected; locally connected; free topological group; Pontryagin's duality; pseudo-open mapping; open mapping; Urysohn space; connectification; connected; locally connected; free topological group; Graev's modification of a group},
language = {eng},
number = {4},
pages = {735-753},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Connectedness and local connectedness of topological groups and extensions},
url = {http://eudml.org/doc/248437},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Alas, Ofelia Teresa
AU - Tkachenko, Mihail G.
AU - Tkachuk, Vladimir Vladimirovich
AU - Wilson, Richard Gordon
TI - Connectedness and local connectedness of topological groups and extensions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 4
SP - 735
EP - 753
AB - 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 $\lbrace 0,1\rbrace ^{\mathfrak {c}}$ of power less than $\mathfrak {c}$, 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.
LA - eng
KW - connected; locally connected; free topological group; Pontryagin's duality; pseudo-open mapping; open mapping; Urysohn space; connectification; connected; locally connected; free topological group; Graev's modification of a group
UR - http://eudml.org/doc/248437
ER -