Connectedness and local connectedness of topological groups and extensions.

*(English)*Zbl 1010.54043The 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.

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.

Reviewer: Ladislav Skula (Brno)