A.G. Chentsov. Ultrafilters and maximal linked systems ... P. 274-292

The structure of ultrafilters of a broadly understood measurable space and of maximal linked systems defined on this space is studied. Bitopological spaces of ultrafilters and maximal linked spaces obtained in both cases by equipping the space with topologies of Wallman and Stone types are considered; the bitopological space of ultrafilters can be considered as a subspace of the bitopological space whose points are maximal linked systems. For an abstract attainability problem with constraints of asymptotic nature, ultrafilters can be used as generalized elements in extension constructions; for the latter case, we present a new implementation that involves the application of linked families of subsets of the set of ordinary solutions in the construction of constraints of asymptotic nature. A natural generalization of the usual “linkedness” is considered, when it is postulated that the intersection of sets of subfamilies of the original family defining the measurable space of cardinality not exceeding a given positive integer is nonempty. For this case, we establish relations connecting ultrafilters and maximal linked systems considered in the specified generalized sense.

Keywords: bitopological space, maximal linked system, topology, ultrafilter

Received November 15, 2019

Revised December 25, 2019

Accepted January 14, 2020

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00410).

Alexander Georgievich Chentsov, Dr. Phys.-Math. Sci, RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: chentsov@imm.uran.ru

REFERENCES

1.   Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov [Theory of Stochastic Processes]. Moscow: Fizmatlit Publ., 2005, 402 p. ISBN: 5-9221-0335-0 .

2.   Dvalishvili B.P. Bitopological spaces: theory, relations with generalized algebraic structures, and applications. Ser. Nort-Holland Mathematics Studies, vol. 199, Amsterdam; Boston; Heidelberg; London; N Y: Elsevier, 2005, 422 p. ISBN: 9780444517937 .

3.   Engelking R. General topology. Ser. Sigma series in pure mathematics, vol. 6. Berlin: Heldermann Verlag, 1989, 535 p. ISBN: 3885380064 . Translated to Russian inder the title Obshchaya topologiya. Moscow: Mir Publ., 1986, 752 p.

4.   Chentsov A.G. Ultrafilters and maximal linked systems: basic properties and topological constructions. Izv. IMI UdGU, 2018, vol. 52, pp. 86–102 (in Russian). doi: 10.20537/2226-3594-2018-52-07 

5.   Chentsov A.G. Filters and ultrafilters in the constructions of attraction sets. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2011, no. 1, pp. 113–142 (in Russian).

6.   Chentsov A.G. Finitely additive measures and extensions of abstract control problems. J. Math. Sci., 2006, vol. 133, no. 2, pp. 1045–1206. doi: 10.1007/s10958-006-0030-0 

7.   Chentsov A.G. Compactifiers in extension constructions for reachability problems with constraints of asymptotic nature. Proc. Steklov Inst. Math., 2017, vol. 296, suppl. 1, pp. 102–118. doi: 10.1134/S0081543817020109 

8.   Bourbaki N. General topology. Chapters 1–4. Berlin; Heidelberg: Springer-Verlag, 1995, 437 p. ISBN: 978-3-642-61701-0 . Translated to Russian under the title Obshchaya topologiya. Osnovnye struktury. Moscow: Nauka Publ., 1968, 272 p.

9.   Chentsov A.G. One representation of the results of action of approximate solutions in a problem with constraints of asymptotic nature. Proc. Steklov Inst. Math., 2012, vol. 276, suppl. 1, pp. S48–S62. doi: 10.1134/S0081543812020046 

10.   Chentsov A.G. On one example of representing the ultrafilter space for an algebra of sets. Trudy Inst. Mat. i Mekh. UrO RAN, 2011, vol. 17, no. 4, pp. 293–311 (in Russian).

11.   Chentsov A.G., Baklanov A.P. On an asymptotic analysis problem related to the construction of an attainability domain. Proc. Steklov Inst. Math., 2015, vol. 291, pp. 279–298. doi: 10.1134/S0081543815080222 

12.   Chentsov A.G., Baklanov A.P., Savenkov I.I. A problem of attainability with constraints of asymptotic nature. Izv. IMI UdGU, 2016, no. 1(47), pp. 54–118 (in Russian).

13.   Chentsov A.G. Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems. Proc. Steklov Inst. Math., 2011, vol. 275, suppl. 1, pp. S12–S39. doi: 10.1134/S0081543811090021 

14.   Chentsov A.G., Pytkeev E.G. Some topological structures of extensions of abstract reachability problems. Proc. Steklov Inst. Math., 2016, vol. 292, suppl. 1, pp. 36–54. doi: 10.1134/S0081543816020048 

15.   Chentsov A.G. Ultrafilters and maximal linked systems. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, vol. 27, no. 3, pp. 365–388 (in Russian). doi: 10.20537/vm170307 

16.   Chentsov A.G. Bitopological spaces of ultrafilters and maximal linked systems. Trudy Inst. Mat. i Mekh. UrO RAN, 2018, vol. 24, no. 1, pp. 257–272 (in Russian). doi: 10.21538/0134-4889-2018-24-1-257-272 

17.   Fedorchuk V.V., Filippov V.V. Obshchaya topologiya. Osnovnye konstruktsii [General topology: Basic constructions]. Moscow: Fizmatlit Publ., 2006, 336 p. ISBN: 5-9221-0618-X

18.   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.

19.   van Mill J. Supercompactness and Wallman spaces. Ser. Math. Center Tracts, no. 85. Amsterdam: Mathematisch Centrum, 1977, 238 p. ISBN: 90-6196-151-3 

20.   Strok M, Szymanski A. Compact metric spaces have binary subbases. Fund. Math., 1975, vol. 89, no. 1, pp. 81–91. doi: 10.4064/fm-89-1-81-91 

21.   Chentsov A.G. Superextension as bitopological space. Izv. IMI UdGU, 2017, vol. 49, pp. 55–79 (in Russian). doi: 10.20537/2226-3594-2017-49-03 

22.   Arkhangel’skii A.V. Compactness. General topology II. Encycl. Math. Sci., 1996, vol. 50, pp. 1–117.

23.   Chentsov A.G. Supercompact spaces of ultrafilters and maximal linked systems. Trudy Inst. Mat. i Mekh. UrO RAN, 2019, vol. 25, no. 2, pp. 240–257 (in Russian). doi: 10.21538/0134-4889-2019-25-2-240-257 

24.   Kuratowski K., Mostowski A. Set theory. ISBN: 9780444534170 . Warszawa: PWN - Polish Scientific Publishers, 1968, 417 p. Translated to Russian under the title Teoriya mnozhestv. Moscow: Mir Publ., 1970, 416 p.

25.   Neveu J. Mathematical foundations of the calculus of probability. San Francisco: Holden-Day, 1965, 223 p. Translated to Russian under the title Matematicheskie osnovy teorii veroyatnostei. Moscow: Mir Publ., 1969, 309 p.

26.   Alexandroff P.S. Einfuhrung in die Mengenlehre und in die allgemeine Topologie [Introduction to set theory and to general topology]. Berlin: VEB Deutscher Verlag der Wissenschaften, 1984, 336 p. Original Russian text published in Aleksandrov P.S. Vvedenie v teoriyu mnozhestv i obshchuyu topologiyu. Moscow: Editorial URSS, 2004, 368 p.

27.   Chentsov A.G. Attraction sets in abstract attainability problems: equivalent representations and basic properties. Russian Math. (Iz. VUZ), 2013, vol. 57, no. 11, pp. 28–44. doi: 10.3103/S1066369X13110030 

28.   Chentsov A.G. The transformation of ultrafilters and their application in constructions of attraction sets. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 3, pp. 85–102. (in Russian)

29.   Chentsov A.G., Morina S.I. Extensions and relaxations, Dordrecht; Boston; London: Kluwer Academic Publ., 2002, 408 p. doi: 10.1007/978-94-017-1527-0 

30.   Chentsov A.G. Some ultrafilter properties connected with extension constructions. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 1, pp. 87–101 (in Russian).

Cite this article as: A.G. Chentsov. Ultrafilters and maximal linked systems, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 1, pp. 274–292.