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Connectedness and local connectedness of topological groups and extensions. (English) Zbl 1010.54043
The first part of this paper is devoted to the question of connectedness of free groups. It is proved (2.3) that if $$X$$ is a connected locally connected space, both the free topological group $$F(X)$$ and the free Abelian topological group $$A(X)$$ are locally connected, and the groups $$F\Gamma (X)$$, $$A\Gamma (X)$$, $$F^*(X)$$ and $$A^*(X)$$ are connected and locally connected. The groups $$F\Gamma (X)$$ and $$\Gamma (X)$$ are Graev’s modifications of the groups [M. I. Graev, Izv. Akad. Nauk SSSR, Ser. Mat. 12, 279-324 (1948; Zbl 0037.01203)]. The groups $$F^*(X)$$ and $$A^*(X)$$ are open normal subgroups of $$F(X)$$ and $$A(X)$$, respectively, such that the quotient groups $$F(X)/F^*(X)$$ and $$A(X)/A^*(X)$$ are isomorphic to the group of rational integers. (All spaces are Hausdorff.)
In the second part an embedding of a (Hausdorff) space into connected or locally connected one is investigated. It is proved (3.8, 3.9), e.g., that any dense in itself subspace of the Sorgenfrey line has a Uryson connectification.
In the conclusion, 12 unsolved problems are presented.

##### MSC:
 54H11 Topological groups (topological aspects) 54D05 Connected and locally connected spaces (general aspects) 54C25 Embedding
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