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Izv. RAN. Ser. Mat., 1995, Volume 59, Issue 4, Pages 15–60 (Mi izv30)  

This article is cited in 5 scientific papers (total in 5 papers)

Extensions of the ring of continuous functions generated by regular, countably-divisible, complete rings of quotients, and their corresponding pre-images

V. K. Zakharov

St. Petersburg State University of Technology and Design

Abstract: In this article we consider metaregular and countably-divisible extensions generated by a regular quotient ring of the ring of continuous functions in the spirit of Fine–Gillman–Lambek. The corresponding pre-images of maximal ideals are considered in connection with these extensions. These pre-images are called small absolutes and a-nonconnected coverings. To characterize these structures a new topological structure is introduced for Aleksandrov spaces with a precovering. In this connection we introduce the notion of a non-connected covering of step type. In the first part of the article we give a characterization of a small absolute as a relatively countably non-connected covering (Theorem 1). We also give a description of the absolute (Theorem 2) and of Aleksandrov pre-images of maximal ideals of Hausdorff–Sierpinski ring extensions (Theorem 3). In the second part we give a characterization of an $a$-non-connected pre-image as an absolutely countably non-connected covering (Theorem 4). Descriptions are also given of Baire and Borel pre-images generated by the classical Baire and Borel measurable extensions (Theorem 5).

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English version:
Izvestiya: Mathematics, 1995, 59:4, 677–720

Bibliographic databases:

MSC: 13B30, 46E25, 54H10
Received: 14.11.1993

Citation: V. K. Zakharov, “Extensions of the ring of continuous functions generated by regular, countably-divisible, complete rings of quotients, and their corresponding pre-images”, Izv. RAN. Ser. Mat., 59:4 (1995), 15–60; Izv. Math., 59:4 (1995), 677–720

Citation in format AMSBIB
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\by V.~K.~Zakharov
\paper Extensions of the ring of continuous functions generated by regular, countably-divisible, complete rings of quotients, and their corresponding pre-images
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\yr 1995
\vol 59
\issue 4
\pages 15--60
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1356349}
\zmath{https://zbmath.org/?q=an:0886.54015}
\transl
\jour Izv. Math.
\yr 1995
\vol 59
\issue 4
\pages 677--720
\crossref{https://doi.org/10.1070/IM1995v059n04ABEH000030}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. K. Zakharov, A. V. Mikhalev, “The problem of general Radon representation for an arbitrary Hausdorff space”, Izv. Math., 63:5 (1999), 881–921  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. V. K. Zakharov, A. V. Mikhalev, “The problem of general Radon representation for an arbitrary Hausdorff space. II”, Izv. Math., 66:6 (2002), 1087–1101  mathnet  crossref  crossref  mathscinet  zmath
    3. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “The Riesz–Radon–Fréchet problem of characterization of integrals”, Russian Math. Surveys, 65:4 (2010), 741–765  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Descriptive spaces and proper classes of functions”, J. Math. Sci., 213:2 (2016), 163–200  mathnet  crossref  mathscinet
    5. V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Postclassical families of functions proper for descriptive and prescriptive spaces”, J. Math. Sci., 221:3 (2017), 360–383  mathnet  crossref  mathscinet
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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