Целью исследования

The aim of the research work is to provide metrization criteria for topological spaces when faced with various covariance functions and to determine the finite dimensions of topological space multiplication and covariance functions moving in broad categories.
The object of the research work. Dimension of a product of a finite number of topological spaces, metrization of topological spaces, extensor and shape properties of spaces of values of spaces under the influence of covariant functors.
Scientific novelty of the research work is as follows:
in the class of paracompact sigma-spaces, it is proved that the dimension of a finite product for measurement in the language of coverings does not exceed the sum of the dimensions of factors;
when X is compact for the covariance function F, a criterion for the metrization of F(X) of compact sets and spatial subsets is defined;
it is proved that subfunctors with finite powers of the functor of probability measures preserve stratifiable spaces and weakly countable-dimensional spaces;
classes of projective-factorial, projectively-inductive closed, and sigma-projectively-inductively closed functors are described, as well as finitely conducting functions are defined in which the equality of shafts of infinite compact sets is invariant;
it is proved that the space of a compact whose diagonal filler is genetically normal at a level greater than or equal to three is metric.
Implementation of research results. Results related to the geometric and topological properties of spaces that are the values ​​of some covariant functors have been used in the following research projects:
The results related to the description of the classes of projective-factorial, projectively-inductive closed and sigma-projectively-inductively closed functors were used in the project OT-F4-69 "Harmonic analysis, power geometry and their applications to problems of mathematical physics" when finding solutions to systems of nonlinear equations for the Cauchy problem (certificate of Samarkand State University dated November 18, 2021, No. 10-4632). The application of these scientific results made it possible to find an infinite number of generalized solutions of the Cauchy problem in the class of Borel measures, for some systems of nonlinear equations with the Fourier transform.
The results of the dimension of the finite product in the class of paracompact sigma-spaces in the language of coverings does not exceed the sum of the dimensions of a finite number of factors were used in a foreign study of mathematical physics to determine the class of isoenergy manifolds of the space of solutions of ordinary differential equations (reference of Samara National Research University named after S.P. Korolev dated December 7, 2022, No. 02-07-3954 4/04). The application of the scientific result made it possible to optimize the properties of solutions of differential equations when matching the boundary conditions for limited and infinite sources of doping of heterostructures.
Results related to subfunctors with finite degrees of the functor of probability measures preserves stratifiable spaces and weakly countable spaces was used in the foreign scientific project "Plasma Technologies" in determining the homotopy dense properties of subspaces (certificate of the Samara State Technical University dated October 25, 2021, No. 01.0202/3277) . The application of the scientific result made it possible to find new topological properties - homotopically dense subsets of classes of isoenergetic manifolds and to determine the stable topological reasons for the evolution of vortices in flows.