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Uniform Completion In Pointfree Topology

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Topological and Algebraic Structures in Fuzzy Sets

Part of the book series: Trends in Logic ((TREN,volume 20))

Abstract

Pointfree topology deals with certain complete lattices, called frames, which may be viewed as abstractly defined lattices of open sets, sufficiently resembling the concrete lattices of this kind that arise from topological spaces to make the treatment of a variety of topological questions possible. It turns out that a remarkable number of topological facts derive from results in this pointfree setting while the proofs of the latter are often more suggestive and transparent than those of their classical counterparts. But there is a deeper aspect of frames which endows them with a very specific significance: various topological spaces classically associated with other entities (such as several types of rings, or Banach spaces, or lattices) are actually the spectra of appropriate frames which themselves require weaker logical foundations for the proofs of their basic properties than those needed for the actual spaces but which can still serve much the same purposes as the spaces in question. In this way, pointfree topology acquires an autonomous rôle and appears as more fundamental than classical topology.

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Authors
  1. B. Banaschewski
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Editors and Affiliations

  1. Department of Mathematics and Statistics, Youngstown State University, 44555-3609, Youngstown, Ohio, USA

    Stephen Ernest Rodabaugh

  2. Institut für Algebra, Stochastik und wissensbasierte mathematische Systeme, Johannes Kepler Universität, 4040, Linz, Austria

    Erich Peter Klement

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Banaschewski, B. (2003). Uniform Completion In Pointfree Topology. In: Rodabaugh, S.E., Klement, E.P. (eds) Topological and Algebraic Structures in Fuzzy Sets. Trends in Logic, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0231-7_2

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  • DOI: https://doi.org/10.1007/978-94-017-0231-7_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6378-6

  • Online ISBN: 978-94-017-0231-7

  • eBook Packages: Springer Book Archive

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