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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, serious
    Publish date - 2020
    Language - english
    COBISS.SI-ID - 18975321

    Link(s):

    https://doi.org/10.1515/taa-2020-0007
    DOI

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