000			04518nam a22005175i 4500
001			u376483
003			SIRSI
005			20160812084415.0
007			cr nn 008mamaa
008			110907s2011 gw | s |||| 0|eng d
020			_a9783642224157 _9978-3-642-22415-7
040			_cMX-MeUAM
050		4	_aQA75.5-76.95
082	0	4	_a004.0151 _223
100	1		_aBridges, Douglas S. _eauthor.
245	1	0	_aApartness and Uniformity _h[recurso electrónico] : _bA Constructive Development / _cby Douglas S. Bridges, Luminita Simona Vîta.
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		_aVîta, Luminita 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	_zLibro electrónico _uhttp://148.231.10.114:2048/login?url=http://link.springer.com/book/10.1007/978-3-642-22415-7
596			_a19
942			_cLIBRO_ELEC
999			_c204363 _d204363