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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, Issue 1, Pages 87–101 (Mi vuu419)  

This article is cited in 12 scientific papers (total in 12 papers)

MATHEMATICS

Some ultrafilter properties connected with extension constructions

A. G. Chentsov

N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620990, Russia

Abstract: General properties of ultrafilters of $\pi$-systems with zero and unit used under extension constructing for abstract attainability problems with the aim of estimation for attraction sets in topological space are considered. Possibilities of employment of the above-mentioned ultrafilters as general elements are considered. Among them, elements admissible with respect to constraints of asymptotic character of the initial problem are selected. Under very general conditions, the goal operator of the given problem extends to the continuous mapping that takes each ultrafilter of $\pi$-system to the limit of corresponding image. The basic attraction set (an asymptotic analog of the attainability domain) is estimated from below by the continuous image of an analogous auxiliary set in the space of ultrafilters. In the particular case of realization of the Stone space (when the used $\pi$-system is an algebra of sets) the above-mentioned estimate is an equality connecting a desired attraction set and an auxiliary one; for the latter a sufficiently simple representation is given. The variant of application (in estimating goals) of the Wallman extension is discussed.

Keywords: attraction set, topology, ultrafilter.

Full text: PDF file (262 kB)
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UDC: 519.6
MSC: 28A33
Received: 15.01.2014

Citation: A. G. Chentsov, “Some ultrafilter properties connected with extension constructions”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 1, 87–101

Citation in format AMSBIB
\Bibitem{Che14}
\by A.~G.~Chentsov
\paper Some ultrafilter properties connected with extension constructions
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2014
\issue 1
\pages 87--101
\mathnet{http://mi.mathnet.ru/vuu419}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. G. Chentsov, “K voprosu o soblyudenii ogranichenii v klasse obobschennykh elementov”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 3, 90–109  mathnet
    2. A. G. Chentsov, “K voprosu o realizatsii elementov prityazheniya v abstraktnykh zadachakh o dostizhimosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:2 (2015), 212–229  mathnet  elib
    3. A. G. Chentsov, “Abstraktnaya zadacha o dostizhimosti: “chisto asimptoticheskaya” versiya”, Tr. IMM UrO RAN, 21, no. 2, 2015, 289–305  mathnet  mathscinet  elib
    4. A. G. Chentsov, “Compactifiers in extension constructions for reachability problems with constraints of asymptotic nature”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 102–118  mathnet  crossref  mathscinet  isi  elib
    5. A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy mnozhestv”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:3 (2017), 365–388  mathnet  crossref  elib
    6. A. G. Chentsov, “Bitopological spaces of ultrafilters and maximal linked systems”, Proc. Steklov Inst. Math. (Suppl.), 305, suppl. 1 (2019), S24–S39  mathnet  crossref  crossref  mathscinet  isi  elib
    7. A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy: osnovnye svoistva i topologicheskie konstruktsii”, Izv. IMI UdGU, 52 (2018), 86–102  mathnet  crossref  elib
    8. A. G. Chentsov, “Maximal linked systems and ultrafilters in abstract attainability problem”, IFAC-PapersOnLine, 51:32 (2018), 239–244  crossref  isi  scopus
    9. A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy: osnovnye sootnosheniya”, Izv. IMI UdGU, 53 (2019), 138–157  mathnet  crossref  elib
    10. A. G. Chentsov, “O superkompaktnosti prostranstva ultrafiltrov s topologiei volmenovskogo tipa”, Izv. IMI UdGU, 54 (2019), 74–101  mathnet  crossref
    11. A. G. Chentsov, “Superkompaktnye prostranstva ultrafiltrov i maksimalnykh stseplennykh sistem”, Tr. IMM UrO RAN, 25, no. 2, 2019, 240–257  mathnet  crossref  elib
    12. A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy”, Tr. IMM UrO RAN, 26, no. 1, 2020, 274–292  mathnet  crossref  elib
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