On the supercompactness of ultrafilter space with the topology of Wallman type

A. G. Chentsov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 74-101.

Résumé

This paper is concerned with ultrafilters and maximal linked systems of widely understood measurable spaces (nonempty sets with $\pi$-systems of its subsets are meant). The sets of ultrafilters and maximal linked systems are transformed to bitopological spaces by applying constructions that (in idea) meet the Wallman and Stone schemes. The focus is on ultrafilter space with topology of Wallman type. Conditions on the initial $\pi$-system for which the given space is supercompact are specified. Concrete classes of (widely understood) measurable spaces are listed for which the above-mentioned conditions are realized. Special attention is also given to one abstract problem of attainability under conditions when the choice of a concrete solution may have the following uncertainty: the set defining constraints can be an arbitrary element of a given nonempty family. The question of the existence of universally realized (in limit) elements in the space of values of the goal operator in our problem is considered. To obtain sufficient solutions, the supercompactness property of the ultrafilter space for special measurable structure is used; this structure is sufficient (under corresponding suppositions) for realization of all variants of constraints on the choice of a usual solution (control).

Détail

Keywords: maximal linked system, topology, ultrafilter.

Bibliographie

[1] Bulinskii A. V., Shiryaev A. N., Theory of random processes, Fizmatlit, M., 2005

[2] Chentsov A. G., “On one example of representing the ultrafilter space for an algebra of sets”, Trudy Instituta Matematiki i Mekhaniki URO RAN, 17, no. 4, 2011, 293–311 (in Russian)

[3] Chentsov A. G., “Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems”, Proceedings of the Steklov Institute of Mathematics, 275, suppl. 1 (2011), 12–39 | DOI | MR | Zbl

[4] Chentsov A. G., “One representation of the results of action of approximate solutions in a problem with constraints of asymptotic nature”, Proceedings of the Steklov Institute of Mathematics, 276, suppl. 1 (2011), 48–62 | DOI | MR | Zbl

[5] Pytkeev E. G., Chentsov A. G., “On the structure of ultrafilters and properties related to convergence in topological spaces”, Proceedings of the Steklov Institute of Mathematics, 289, suppl. 1 (2014), 164–181 | DOI | MR

[6] Chentsov A. G., Pytkeev E. G., “Some topological structures of extensions of abstract reachability problems”, Proceedings of the Steklov Institute of Mathematics, 292, suppl. 1 (2016), 36–54 | DOI | MR

[7] Gryzlov A. A., Bastrykov E. S., Golovastov R. A., “About points of compactification of $N$”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2010, no. 3, 10–17 (in Russian) | DOI

[8] Gryzlov A. A., Golovastov R. A., “The Stone spaces of Boolean algebras”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, no. 1, 11–16 (in Russian) | DOI

[9] Golovastov R. A., “About Stone space of one Boolean algebra”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, no. 3, 19–24 (in Russian) | DOI

[10] Fedorchuk V. V., Filippov V. V., General topology. Base constructions, Fizmatlit, M., 2006

[11] de Groot J., “Superextensions and supercompactness”, Proc. I. Intern. Symp. on extension theory of topological structures and its applications, VEB Deutscher Verlag Wis., Berlin, 1969, 89–90 | MR

[12] van Mill J., Supercompactness and Wallman spaces, Mathematisch Centrum, Amsterdam, 1977 | MR | Zbl

[13] Strok M., Szymański A., “Compact metric spaces have binary bases”, Fund. Math., 89:1 (1975), 81–91 | DOI | MR | Zbl

[14] Chentsov A. G., “Superrasshirenie kak bitopologicheskoe prostranstvo”, Izvestiya Instituta matematiki i informatiki Udmurtskogo gosudarstvennogo universiteta, 49 (2017), 55–79 | DOI | Zbl

[15] Chentsov A. G., “Ultrafilters and maximal linked systems”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 27:3 (2017), 365–388 (in Russian) | DOI | Zbl

[16] Chentsov A. G., “Some representations connected with ultrafilters and maximal linked systems”, Ural Mathematical Journal, 3:2 (2017), 100–121 | DOI | MR

[17] Chentsov A. G., “Bitopological spaces of ultrafilters and maximal linked systems”, Trudy Instituta Matematiki i Mekhaniki URO RAN, 24, no. 1, 2018, 257–272 (in Russian) | DOI

[18] Chentsov A. G., “Ultrafilters and maximal linked systems: basic properties and topological constructions”, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 52 (2018), 86–102 (in Russian) | DOI

[19] Arkhangelskii A. V., “Compactness”, General Topology II, Encyclopaedia Math. Sci., 50, Springer-Verlag, Berlin, 1996, 1–117 | MR

[20] Dieudonné J., Foundations of modern analysis, Academic Press Inc., New York, 1960 | MR | Zbl

[21] Kuratowski K., Mostowski A., Set theory, North-Holland, Amsterdam, 1967 | MR

[22] Chentsov A. G., Elements of a finitely additive measure theory, v. I, USTU–UPI, Yekaterinburg, 2009

[23] Aleksandrov P. S., Introduction to set theory and general topology, Editorial URSS, M., 2004

[24] Engelking R., General topology, Państwowe Wydawnictwo Naukowe, Warszawa, 1985

[25] Chentsov A. G., Elements of a finitely additive measure theory, v. II, USTU–UPI, Yekaterinburg, 2010

[26] Chentsov A. G., “The transformation of ultrafilters and their application in constructions of attraction sets”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, no. 3, 85–102 (in Russian) | DOI

[27] Chentsov A. G., Morina S. I., Extensions and relaxations, Springer Netherlands, Dordrecht, 2002 | DOI | MR

[28] Bogachev V. I., Weak convergence of measures, Institute of Computer Science, M.–Izhevsk, 2016

[29] Neve Zh., Mathematical basis of probabilities theory, Mir, M., 1969

[30] Warga J., Optimal control of differential and functional equations, Academic Press, New York, 1972 | DOI | MR | Zbl

[31] Gamkrelidze R. V., Foundations of optimal control, Tbilisi University, Tbilisi, 1975

[32] Krasovskii N. N., Theory of motion control, Nauka, M., 1968

[33] Krasovskii N. N., Subbotin A. I., Game-theoretical control problems, Springer, New York, 1988 | MR | Zbl

[34] Subbotin A. I., Chentsov A. G., Guaranteed optimization in control problems, Nauka, M., 1981

[35] Chentsov A. G., Finitely additive measures and relaxations of extremal problems, Springer US, New York, 1996 | MR

[36] Chentsov A. G., Asymptotic attainability, Springer Netherlands, Dordrecht, 1997 | DOI | MR

[37] Chentsov A. G., “Compactifiers in extension constructions for reachability problems with constraints of asymptotic nature”, Proceedings of the Steklov Institute of Mathematics, 296, suppl. 1 (2017), 102–118 | DOI | MR

[38] Chentsov A. G., “Filters and ultrafilters in the constructions of attraction sets”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2011, no. 1, 113–142 (in Russian) | DOI

[39] Chentsov A. G., “Some ultrafilter properties connected with extension constructions”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, no. 1, 87–101 (in Russian) | DOI

[40] Burbaki N., General topology, Nauka, M., 1968

[41] Kelley J. L., General topology, Springer, 1955 | MR

[42] Chentsov A. G., “The mappings and ultrafilter-based transformations”, Trudy Instituta Matematiki i Mekhaniki URO RAN, 18, no. 4, 2012, 298–314