# Connectedness and local connectedness of topological groups and extensions

Ofelia Teresa Alas; Mihail G. Tkachenko; Vladimir Vladimirovich Tkachuk; Richard Gordon Wilson

Commentationes Mathematicae Universitatis Carolinae (1999)

- Volume: 40, Issue: 4, page 735-753
- ISSN: 0010-2628

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topAlas, 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 -

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