# Article

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Keywords:
connected; locally connected; free topological group; Pontryagin's duality; pseudo-open mapping; open mapping; Urysohn space; connectification
Summary:
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.
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