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001 978-3-642-22415-7
003 DE-He213
005 20140220083808.0
007 cr nn 008mamaa
008 110907s2011 gw | s |||| 0|eng d
020 _a9783642224157
_9978-3-642-22415-7
024 7 _a10.1007/978-3-642-22415-7
_2doi
050 4 _aQA75.5-76.95
072 7 _aUY
_2bicssc
072 7 _aUYA
_2bicssc
072 7 _aCOM014000
_2bisacsh
072 7 _aCOM031000
_2bisacsh
082 0 4 _a004.0151
_223
100 1 _aBridges, Douglas S.
_eauthor.
245 1 0 _aApartness and Uniformity
_h[electronic resource] :
_bA Constructive Development /
_cby Douglas S. Bridges, Luminiţa Simona Vîţă.
264 1 _aBerlin, Heidelberg :
_bSpringer Berlin Heidelberg,
_c2011.
300 _aXIV, 198 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aTheory and Applications of Computability, In cooperation with the association Computability in Europe,
_x2190-619X
505 0 _aThe Constructive Framework -- Point-Set Apartness -- Apartness Between Sets -- Postlude: Paths to Topology -- References -- Index.
520 _aThe 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 relation between apartness spaces and formal topology. Largely an exposition of the authors' own research, this is the first book dealing with the apartness approach to constructive topology, and is a valuable addition to the literature on constructive mathematics and on topology in computer science. It is aimed at graduate students and advanced researchers in theoretical computer science, mathematics, and logic who are interested in constructive/algorithmic aspects of topology.
650 0 _aComputer science.
650 0 _aInformation theory.
650 0 _aGlobal analysis (Mathematics).
650 0 _aLogic, Symbolic and mathematical.
650 0 _aTopology.
650 1 4 _aComputer Science.
650 2 4 _aTheory of Computation.
650 2 4 _aMathematics of Computing.
650 2 4 _aTopology.
650 2 4 _aAnalysis.
650 2 4 _aMathematical Logic and Foundations.
700 1 _aVîţă, Luminiţa Simona.
_eauthor.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783642224140
830 0 _aTheory and Applications of Computability, In cooperation with the association Computability in Europe,
_x2190-619X
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-642-22415-7
912 _aZDB-2-SCS
999 _c108190
_d108190