O.T. Alas, M.G. Tka\v cenko, V.V. Tkachuk, R.G. Wilson
Connectedness and local connectedness of topological groups and extensions

Comment.Math.Univ.Carolinae 40,4 (1999) 735-753.

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. \par 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. \par 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 $D$. \par 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.

Keywords: connected, locally connected, free topological group, Pontryagin's duality, pseudo-open mapping, open mapping, Urysohn space, connectification
AMS Subject Classification: Primary 54H11, 54C10, 22A05, 54D06; Secondary 54D25, 54C25