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Positive answers to Koch's problem in special cases [Elektronski vir]Banakh, Taras, 1968- ...A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for ... more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch's problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid is a compact topological group if and only if ▫$S$▫ is a submonoid of a quasitopological group if and only if ▫$S$▫ has open shifts if and only if ▫$S$▫ is non-viscous in the sense of Averbukh. The last condition means that any neighborhood ▫$U$▫ of the identity ▫$1$▫ of ▫$S$▫ and for any element ▫$a\in S$▫ there exists a neighborhood ▫$V$▫ of ▫$a$▫ such that any element ▫$x\in S$▫ with ▫$(xV\cup Vx)\cap V\ne\emptyset$▫ belongs to the neighborhood ▫$U$▫ of 1.Source: Topological algebra and its applications [Elektronski vir]. - ISSN 2299-3231 (Vol. 8, iss. 1, Jan. 2020, str. 76-87)Type of material - e-article ; adult, seriousPublish date - 2020Language - englishCOBISS.SI-ID - 18975321
Author
Banakh, Taras, 1968- |
Bardyla, Serhii |
Guran, Igor |
Gutik, Oleg |
Ravsky, O. V.
Topics
Koch's problem |
monothetic semigroup |
non-viscous monoid |
topological semigroup |
semitopological semigroup |
cancellative semigroup |
locally compact semigroup |
compact-like semigroup |
sequentially compact semigroup |
countably compact semigroup |
feebly compact semigroup |
countably pracompact semigroup |
Tkachenko-Tomita group
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Banakh, Taras, 1968- | 31193 |
Bardyla, Serhii | ![]() |
Guran, Igor | ![]() |
Gutik, Oleg | ![]() |
Ravsky, O. V. | ![]() |
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