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
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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Izvestiya: Mathematics, 1995, 59:4, 677–720
MSC: 13B30, 46E25, 54H10
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
\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.
\jour Izv. Math.
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V. K. Zakharov, A. V. Mikhalev, “The problem of general Radon representation for an arbitrary Hausdorff space”, Izv. Math., 63:5 (1999), 881–921
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
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
V. K. Zakharov, A. V. Mikhalev, T. V. Rodionov, “Descriptive spaces and proper classes of functions”, J. Math. Sci., 213:2 (2016), 163–200
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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