rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Apartness and Uniformity
    sub title: A Constructive Development
  name:
    name part: Bridges, Douglas S.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: Vîţă, Luminiţa Simona.
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: gw
    date issued: 2011
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XIV, 198 p. online resource.
  abstract: The theory presented in this book is developed constructively, is based on a few axioms encapsulating the notion of objects (points and sets) being apart, and encompasses both point-set topology and the theory of uniform spaces. While the classical-logic-based theory of proximity spaces provides some guidance for the theory of apartness, the notion of nearness/proximity does not embody enough algorithmic information for a deep constructive development. The use of constructive (intuitionistic) logic in this book requires much more technical ingenuity than one finds in classical proximity theory -- algorithmic information does not come cheaply -- but it often reveals distinctions that are rendered invisible by classical logic. In the first chapter the authors outline informal constructive logic and set theory, and, briefly, the basic notions and notations for metric and topological spaces. In the second they introduce axioms for a point-set apartness and then explore some of the consequences of those axioms. In particular, they examine a natural topology associated with an apartness space, and relations between various types of continuity of mappings. In the third chapter the authors extend the notion of point-set (pre-)apartness axiomatically to one of (pre-)apartness between subsets of an inhabited set. They then provide axioms for a quasiuniform space, perhaps the most important type of set-set apartness space. Quasiuniform spaces play a major role in the remainder of the chapter, which covers such topics as the connection between uniform and strong continuity (arguably the most technically difficult part of the book), apartness and convergence in function spaces, types of completeness, and neat compactness. Each chapter has a Notes section, in which are found comments on the definitions, results, and proofs, as well as occasional pointers to future work. The book ends with a Postlude that refers to other constructive approaches to topology, with emphasis on the re…
  table of contents: The Constructive Framework -- Point-Set Apartness -- Apartness Between Sets -- Postlude: Paths to Topology -- References -- Index.
  note: by Douglas S. Bridges, Luminiţa Simona Vîţă.
  subject:
    topic: Computer science
  subject:
    topic: Computers
  subject:
    topic: Computer science
    topic: Mathematics
  subject:
    topic: Mathematical analysis
  subject:
    topic: Analysis (Mathematics)
  subject:
    topic: Mathematical logic
  subject:
    topic: Topology
  subject:
    topic: Computer Science
  subject:
    topic: Theory of Computation
  subject:
    topic: Mathematics of Computing
  subject:
    topic: Topology
  subject:
    topic: Analysis
  subject:
    topic: Mathematical Logic and Foundations
  classification: QA75.5-76.95
  classification: 004.0151
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: Theory and Applications of Computability, In cooperation with the association Computability in Europe
  identifier: 9783642224157
  identifier: http://dx.doi.org/10.1007/978-3-642-22415-7
  record info:
    record creation date: 110907
    record change date: 20151111071404.0
    record identifier: 978-3-642-22415-7