Free products of topological groups and M-equivalence

In the paper we apply free products of topological groups for investigating the M-equivalence of Tychonoff spaces which are infinite disjoint sums of its subspaces. The main result is the following:Let X=∞i=1⊕Xᵢ and for every i∈ N there exists a topological group Gᵢ such that F(Xᵢ₊₁) is topologically isomorphic to the free product F(Xᵢ) and Gᵢ.Let \ nₖ \ₖ₌₁∞ be an~increasing sequence of the natural numbers and let X̃=∞k=1⊕X_nₖ. Then XM~ X̃.This theorem give us many examples of topological spaces with topologically isomorphic free topological groups. We use obtained results for constructing M-equivalent pairs and M-equivalent bundles of such spaces.