О размерностях финитной аппроксимации и функторах, сохраняющих ε-сети
// Труды КарНЦ РАН. No 4. Сер. Математическое моделирование и информационные технологии. 2022. C. 30-36
A.V. Ivanov. On dimensions of finite approximation and functors preserving ε-nets // Transactions of Karelian Research Centre of Russian Academy of Science. No 4. Mathematical Modeling and Information Technologies. 2022. Pp. 30-36
Keywords: seminormal functor; metrizable functor; dimension of finite approximation; box dimension; quantization dimension
The dimension of finite approximation dimFξ is defined for any point ξ of the space F(X), where F is a metrizable seminormal functor and X is a metric compactum. Such points can represent closed subsets, probability measures, maximal linked systems of closed sets, etc. The analysis shows that for many functorial constructions of general topology (exponential functor, probability measures functor, superextension functor, etc.) the dimension of finite approximation dimF ξ does not exceed the box dimension dimB of the support supp(ξ) of this point. In this connection, the problem naturally arises of how to describe the class of functors for which the indicated restriction on the dimensionality of the finite approximation holds. This paper introduces the notion of a metrizable functor that preserves ε-nets. The condition that ε-nets be preserved by the functor F proved to be sufficient for the inequality dimFξ <= dimB(supp(ξ)) for any point ξ ∈ F(X). It is proved that a number of well-known metrizable functors preserve ε-nets.
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Last modified: June 27, 2022