Email this article Maximal linked systems and ultrafilters: main representations and topological properties
- Authors: Chentsov A.G.1,2
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Affiliations:
- N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences
- Ural Federal University named after the first President of Russia B.N. Yeltsin
- Issue: Vol 25, No 129 (2020)
- Pages: 68-84
- Section: Articles
- URL: https://journals.rcsi.science/2686-9667/article/view/295067
- DOI: https://doi.org/10.20310/2686-9667-2020-25-129-68-84
- ID: 295067
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Abstract
Questions connected with representation of the ultrafilter (UF) set for widely understood measurable space are investigated; this set is considered as a subspace of bitopological space of maximal linked systems (MLS) under equipment with topologies of Wallman and Stone types (measurable structure is defined as a π -system with “zero” and “unit”). Analogous representations connected with generalized variant of cohesion is considered also; in this variant, for corresponding set family, it is postulated the nonemptyness of intersection for finite subfamilies with power not exceeding given. Conditions of identification of UF and MLS (in the above-mentioned generalized sense) are investigated. Constructions reducing to bitopological spaces with points in the form of MLS and n -supercompactness property generalizing the “usual” supercompactness are considered. Finally, some characteristic properties of MLS and their corollaries connected with the MLS contraction to a smaller π -system are being studied. The case of algebras of sets is selected separately.
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About the authors
Aleksandr G. Chentsov
N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences; Ural Federal University named after the first President of Russia B.N. Yeltsin
Email: chentsov@imm.uran.ru
Doctor of Physics and Mathematics, Corresponding Member of the Russian Academy of Sciences, Chief Researcher 16 S. Kovalevskaya St., Yekaterinburg 620108, Russian Federation; 19 Mira St., Yekaterinburg 620002, Russian Federation
References
- А. Г. Ченцов, “Компактификаторы в конструкциях расширений задач о достижимости с ограничениями асимптотического характера”, Тр. ИММ УрО РАН, 22, 2016, 294-309.
- А.Г. Ченцов, “Фильтры и ультрафильтры в конструкциях множеств притяжения”, Вестник Удмуртского университета. Математика. Механика. Компьютерные науки, 2011, №1, 113-142.
- Р. Энгелькинг, Общая топология, Мир, М., 1986, 752 с.
- B.P. Dvalishvili, Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures, and Applications, Mathematics studies, Nort-Holland, 2005, 422 pp.
- В.В. Федорчук, В.В. Филиппов, Общая топология. Основные конструкции, Физматлит, М., 2006, 336 с.
- J. de Groot, “Superextensions and supercompactness”, Extension Theory of Topological Structures and its Applications, I International Symposium “Extension Theory of Topological Structures and its Applications” (Berlin, 1969), VEB Deutscher Verlag Wis., Berlin, 1969, 89-90.
- J.van. Mill, Supercompactness and Wallman Spaces, Mathematisch Centrum, Amsterdam, 1977, 238 pp.
- M. Strok, A. Szymanski, “Compact metric spaces have binary subbases”, Fund. Math., 89:1 (1975), 81-91.
- А.В. Булинский, А.Н. Ширяев, Теория случайных процессов, Физматлит, М., 2005, 402 с.
- А.Г. Ченцов, “Суперкомпактные пространства ультрафильтров и максимальных сцепленныхсистем”, Тр. ИММ УрО РАН, 25, 2019, 240-257.
- К. Куратовский, А. Мостовский, Теория множеств, Мир, М., 1970, 416 с.
- А.Г. Ченцов, “Битопологические пространства ультрафильтров и максимальных сцепленных систем”, Тр. ИММ УрО РАН, 24, 2018, 257-272.
- А.В. Архангельский, “Компактность”, Общая топология - 2, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 50, ВИНИТИ, М., 1989, 5-128.
- Ж. Неве, Математические основы теории вероятностей, Мир, М., 1969, 310 с.
- А.Г. Ченцов, “Ультрафильтры и максимальные сцепленные системы множеств”, Вестн. Удмуртск. ун-та. Матем. Мех. Компьют. науки, 27:3 (2017), 365-388.
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