Connectedness and local connectedness of topological groups and extensions

Alas, O. T. ; Tkačenko, M. G. ; Tkachuk, V. V. ; Wilson, R. G.

Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 4, pp. 735-753.

Résumé

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.

Détail

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