Markov equivalence of the bundles of tychonoff spaces and isomorphisms of the groups of continuous mappings

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Pyrch N. M. № 2 (59) 72-79 Image Image

In the paper, we consider the equivalence of the bundles of the Tychonoff spaces. For a system of topological subspaces of topological space , system of topological subspaces of topological space and a functor of topological algebra we say that the bundle is F-equivalent to the bundle if there exists topological isomorphism between free topological algebras and mapping subalgebra generated by subspace to subalgebra generated by subspace . The problem of the isomorphic classification of the bundles of Tychonoff spaces is more complicated than the isomorphic classification of Tychonoff spaces. To solve this problem and obtain the classification for some special types of spaces and their subspace, we need general methods for constructing such bundles (sufficient conditions) and some properties of the bundles preserved by this relations. (necessity conditions). Such properties characterizing position of the bundle of spaces in Tychonoff space are called M-invariants. Using the results proved in this paper, we can obtain the series of such necessity conditions for the equivalence of the bundles of Tychonoff spaces. For every such isomorphism for every topological group there exists topological isomorphism be the groups of continuous mappings into G in the topology of pointwise convergence. We consider the series of subgroups in which depend on the subspaces and the same subgroups in such that .. In the paper, we define at first as a subsets in , then we prove that all defined are subgroups in and then we check that . All the results are obtained for the relation of M-equivalence, but if the group is abelian, these results are valid for A-equivalence. If G is a group of real numbers with the topology generated by Euclidian metrics, we can obtain the linear homeomorphism between linear spaces and the results should be concerned with free locally convex spaces and the relation of L-equivalence. Under some conditions we can transform for the spaces of the continuous bounded functions.

Keywords: free topological group, M-equivalence, bundle of Tychonoff spaces, the space of continuous mappings, topology of pointwise convergence.

doi: 10.32403/1998-6912-2019-2-59-72-79

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