Abstract

Throughout this paper, all topological groups are assumed to be topological differential manifolds and algebraically free, our aim in this paper is to prove the open problems number (7) and (8). Which are introduced by (Guran, I, 1998). In many cases of spaces and under a suitable conditions. therefore, we denote by I(X) and I(Y) to be a free topological groups over a topological spaces X and Y respectively where X and Y are assumed to be a non- empty sub manifolds Which are also a closed sub sets, and P is a classes of topological spaces, as a regular, normal, Tychonoff, lindelöf, separable connected, compact and Zero- dimensional space, and we have tried to use a hereditary properties and others of these spaces, so we can prove the open problems in these cases and we have many results showed in this paper.

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