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Home » Issues » Archive of Issues » 2020 - 3 issue » pp. 444-467

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Russia Yekaterinburg
Year
2020
Volume
30
Issue
3
Pages
444-467
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Section Mathematics
Title Filters and linked families of sets
Author(-s) Chentsov A.G.ab
Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb
Abstract Properties of ultrafilters (u/f) and maximal linked systems (MLS) on the widely understood measurable space (MS) and representations of linked (not necessarily maximal) families and filters on this MS are investigated. Conditions realizing maximality of linked families (systems) and natural representations for bitopological spaces (BTS) of u/f and MLS are established. Equipments of sets of linked families and filters corresponding to Wallman and Stone schemes are studied; the connection of these equipments with analogous equipments (with topologies) for u/f and MLS leading to above-mentioned BTS is studied too. Properties of linked family products for two (widely understood) MS are investigated. It is shown that MLS on the $\pi$-system product (that is, on the family of “measurable” rectangles) are limited to products of corresponding MLS on initial spaces.
Keywords maximal linked system, family of sets, topology, ultrafilter
UDC 519.6
MSC 93C83
DOI 10.35634/vm200307
Received 3 August 2020
Language Russian
Citation Chentsov A.G. Filters and linked families of sets, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 3, pp. 444-467.
References
  1. de Groot J. Superextensions and supercompactness, Proc. I. Intern. Symp. on extension theory of topological structures and its applications, Berlin: VEB Deutscher Verlag Wis., 1969, pp. 89-90.
  2. van Mill J. Supercompactness and Wallman spaces, Amsterdam: Mathematisch Centrum, 1977.
  3. Strok M., Szymański A. Compact metric spaces have binary bases, Fundamenta Mathematicae, 1975, vol. 89, no. 1, pp. 81-91. https://doi.org/10.4064/fm-89-1-81-91
  4. Fedorchuk V.V., Filippov V.V. Obshchaya topologiya. Osnovnye konstruktsii (General topology. Base constructions), Moscow: Fizmatlit, 2006.
  5. Chentsov A.G. Ultrafilters and maximal linked systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 3, pp. 365-388 (in Russian). https://doi.org/10.20537/vm170307
  6. Chentsov A.G. To a question on the supercompactness of ultrafilter spaces, Ural Math. J., 2019, vol. 5, no. 1, pp. 31-47. https://doi.org/10.15826/umj.2019.1.004
  7. Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov (Theory of random processes), Moscow: Fizmatlit, 2005.
  8. Dvalishvili B.P. Bitopological spaces: theory, relations with generalized algebraic structures, and applications, Amsterdam: North-Holland, 2005.
  9. Chentsov A.G. Supercompact spaces of ultrafilters and maximal linked systems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 2, pp. 240-257 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-2-240-257
  10. Gryzlov A.A. On convergent sequences and copies of $\beta{N}$ in one compactification of $N$, XI Prague Symposium on General Topology, Prague, Czech. Rep., 2011, p. 29.
  11. Gryzlov A.A., Bastrykov E.S., Golovastov R.A. About points of compactification of $N$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2010, issue 3, pp. 10-17 (in Russian). https://doi.org/10.20537/vm100302
  12. Gryzlov A.A., Golovastov R.A. The Stone spaces of Boolean algebras, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 1, pp. 11-16 (in Russian). https://doi.org/10.20537/vm130102
  13. Chentsov A.G. Bitopological spaces of ultrafilters and maximal linked systems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, vol. 24, no. 1, pp. 257-272 (in Russian). https://doi.org/10.21538/0134-4889-2018-24-1-257-272
  14. Kuratowski K., Mostowski A. Set theory, Amsterdam: North-Holland, 1967.
    Translated under the title Teoriya mnozhestv, Moscow: Mir, 1970.
  15. Aleksandrov P.S. Vvedenie v teoriyu mnozhestv i obshchuyu topologiyu (Introduction to set theory and general topology), Moscow: Editorial URSS, 2004.
  16. Neve Zh. Matematicheskie osnovy teorii veroyatnostei (Mathematical foundations of probability theory), Moscow: Mir, 1969.
  17. Chentsov A.G. Ultrafilters and maximal linked systems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 1, pp. 274-292 (in Russian). https://doi.org/10.21538/0134-4889-2020-26-1-274-292
  18. Chentsov A.G. Attraction sets in abstract attainability problems: Equivalent representations and basic properties, Russian Mathematics, 2013, vol. 57, no. 11, pp. 28-44. https://doi.org/10.3103/S1066369X13110030
  19. Engelking R. General topology, Warszawa: Państwowe Wydawnictwo Naukowe, 1985.
    Translated under the title Obshchaya topologiya, Moscow: Mir, 1986.
  20. Chentsov A.G. Ultrafilters and maximal linked systems: basic properties and topological constructions, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2018, vol. 52, pp. 86-102 (in Russian). https://doi.org/10.20537/2226-3594-2018-52-07
  21. Chentsov A.G. On the supercompactness of ultrafilter space with the topology of Walman type, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2019, vol. 54, pp. 74-101. https://doi.org/10.20537/2226-3594-2019-54-07
  22. Chentsov A.G. Some properties of ultrafilters of widely understood measurable spaces, Doklady Akademii Nauk, 2019, vol. 486, no. 1, pp. 24-29.
  23. Chentsov A.G. Maximal linked systems and ultrafilters: basic rerpesentations and topological properties, Vestnik Rossiiskikh Universitetov. Matematika, 2020, vol. 25, no. 129, pp. 68-84. https://doi.org/10.20310/2686-9667-2020-25-129-68-84
  24. Chentsov A.G. Elementy konechno-additivnoi teorii mery, I (Elements of finitely additive measure theory, I), Yekaterinburg: USTU-UPI, 2008.
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