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

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

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
\Bibitem{Zak95} \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 \jour Izv. RAN. Ser. Mat. \yr 1995 \vol 59 \issue 4 \pages 15--60 \mathnet{http://mi.mathnet.ru/izv30} \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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000169556400003} 

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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
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
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
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
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
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