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# Topic: Neighbourhood topology

 Neighbourhood (mathematics) - Wikipedia, the free encyclopedia In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points. Neighbourhoods in more than one dimension are generally chosen equivalent to Eucledian metrics for their symmetry and readability. en.wikipedia.org /wiki/Neighbourhood_%28mathematics%29   (679 words)

 Encyclopedia: Algebraic topology The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants, by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism of spaces. In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions such as simplicial complexes. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). www.nationmaster.com /encyclopedia/Algebraic-topology   (2269 words)

 neighbourhood (topology)   (Site not responding. Last check: 2007-11-07) The topology generated by a base is the smallest topology containing the base elements; this topology consists of all unions of elements of the base. The topology generated by a subbase is the smallest topology containing the subbase elements; this topology consists of all finite intersections of unions of elements of the subbase. The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest topology on the set which makes all the functions continuous. www.yourencyclopedia.net /Neighbourhood_(topology).html   (2549 words)

 Encyclopedia: Topological space The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. www.nationmaster.com /encyclopedia/Topological-space   (6141 words)

 Neighbourhood (topology)   (Site not responding. Last check: 2007-11-07) The topology T is the smallest topology on Xcontaining B and is said to be generated by B. Compact-open topology The compact-open topology on the set C(X,Y) of all continuous maps between two spaces X and Y is defined as follows: given a compact subsetK of X and an open subset U of Y, let V(K, U) denote the set ofall maps f in C(X, Y) such that f(K) is contained in U. The weak topology on a set, with respect to a collection of functions from that set into topologicalspaces, is the coarsest topology on the set which makes all the functions continuous. www.therfcc.org /neighbourhood-topology--86494.html   (3583 words)

 Station Information - Neighbourhood A neighbourhood (in American English, neighborhood) is a geographically localised community located within a larger city or suburb. The residents of a given neighbourhood are called neighbours (or neighbors), although this term may also be used across much larger distances in rural areas. The boroughs of New York City and Greater London are intermediate in size between the neighbourhoods that comprise them on the one hand and the entire city on the other. www.stationinformation.com /encyclopedia/n/ne/neighbourhood.html   (212 words)

 T1 space - Wikipedia, the free encyclopedia Since y is in the closure of {x}, this would force y not to be in U, contradicting the fact that U is a neighbourhood of y. But it is clear that X is a neighbourhood of y that does not contain infinitely many points of S. The Zariski topology on an algebraic variety is T en.wikipedia.org /wiki/T1_topology   (835 words)

 An Atlas of Cyberspaces - Topology Maps This graph is a snapshot of the local gnutella peer network in my neighbourhood created using the mapping functions of the Gnucleus client. A topology cybermap of search results from the AltaVista search engine using their LiveTopics system. The link map was created by Eva Ekeblad and her paper "The emergence and decay of multilogue: self-regulation of a scholarly mailinglist." give more details on this research. www.cybergeography.org /atlas/more_topology.html   (567 words)

 Neighbourhood   (Site not responding. Last check: 2007-11-07) A neighbourhood (in British English) orneighborhood (in American English) is ageographically localised community located within a larger city or suburb. The residents of a given neighbourhood are calledneighbours (or neighbors), although this term may also be used across muchlarger distances in rural areas. In Canada and the UnitedStates, neighbourhoods are often given official or semi-official status through neighbourhoodassociations, or Blockwatch in Canada. www.therfcc.org /neighbourhood-86039.html   (247 words)

 Talk:Topological space - InformationBlast But the relation between sets and points (an open set or neighbourhood containing a point is"close to the point") does (this way of thinking is basically the equivalent definition in terms of neighbourhood systems). Roughly speaking, a topology is a way of specifying the concept of "nearness"; an open set is "near" each of its points. An open set is a member of a topology, and a topology can be defined in any number of equivalent ways (the "neighbourhood" or"nearness" relation between 2 other ones). www.informationblast.com /Talk:Topological_space.html   (2144 words)

 Neighbourhood A neighbourhood (in British English) or neighborhood (in American English) is a geographically localised community located within a larger city or suburb. In some other places the equivalent organisation is the parish, though a parish may have several neighbourhoods within it depending on the area. In the People's Republic of China, the term is generally used for the urban administrative unit usually found immediately below the district level, although an intermediate, subdistrict level exists in some cities. www.brainyencyclopedia.com /encyclopedia/n/ne/neighbourhood.html   (289 words)

 Science Fair Projects - Topology glossary Trivially the neighbourhood system for a point is also a neighbourhood basis for the point. More generally, this remains true whenever the topology is defined by a translation invariant metric or pseudometric. Every neighbourhood system for a non empty set A is a filter called the neighbourhood filter for A. www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Local_base   (381 words)

 Algebraic Topology: Topology The topology on A defined by F is the weakest topology (i.e., the smallest collection OA) for which all these functions become continuous. The topology on B defined by F is the strongest topology (i.e., the largest collection OB) for which all these functions become continuous. A topological space is called metric when there is a distance function determining the topology (i.e., open balls for the metric are open sets, and conversely, if a point x lies in an open set U then for some positive e the ball with radius e around x is contained in U. www.win.tue.nl /~aeb/at/algtop-2.html   (1509 words)

 Topology history The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common. Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler. A second way in which topology developed was through the generalisation of the ideas of convergence. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Topology_in_mathematics.html   (1455 words)

 Neighbourhood   (Site not responding. Last check: 2007-11-07) Then T = is a topology on S, and the resulting space is called Sierpinski space. Specialty definitions using "neighbourhood": Abel-beth-maachah, Aston dark space ♦ brave west wind ♦ Camarina ♦ Dun Cow ♦ electronic cottage, Ezel ♦ Hundred Miles, Husband's Boat ♦ Long Meg of Westminster ♦ Martha, Mazikeen ♦ photoconductor diode, photodiode, Public-house Signs ♦ Red Sea, Passage of, Ribbon Dodge ♦ Shalim, Land of ♦ telework. "Neighbourhood" is used about 1,455 times out of a sample of 100 million words spoken or written in English. www.websters-online-dictionary.org /ne/neighbourhood.html   (3661 words)

 Talk:Topological space - Information   (Site not responding. Last check: 2007-11-07) But the relation between sets and points (an open set or neighbourhood containing a point is "close to the point") does (this way of thinking is basically the equivalent definition in terms of neighbourhood systems). If the intuitive meaning of "near" is too confusing, then maybe it shouldn't be there, but it does have a precise legit meaning in terms of neighbourhood systems. An open set is a member of a topology, and a topology can be defined in any number of equivalent ways (the "neighbourhood" or "nearness" relation between 2 other ones). www.book-spot.co.uk /index.php/Talk:Topological_space   (1635 words)

 [No title]   (Site not responding. Last check: 2007-11-07) topologies are defined by the collection of open sets). Subspaces: defn of subspace topology, closed sets in subspace topology, continuity of inclusion, continuity of maps into A if and only if continuous as maps into X, this last property characterises the subspace topology (ex!). What subbases are good for: sufficient to check continuity on subbasic open sets; easy to build topologies, eg, the weak topology generated by some maps. www.maths.bath.ac.uk /~feb/math0055/diary   (553 words)

 Topology glossary   (Site not responding. Last check: 2007-11-07) See basic set theory, axiomatic set theory, and function for definitions concerning sets and functions. The Kuratowski closure axioms can then be used to define a topology on X by declaring the closed sets to be the fixed pointss of this operator, i.e. (Note that the neighbourhood itself need not be open.) A neighbourhood of a point x is a neighbourhood of the singleton set {x}. www.sciencedaily.com /encyclopedia/topology_glossary   (3712 words)

 Science Fair Projects - First-countable space In topology, a first-countable space is a topological space satisfying the "first axiom of countability". To see this, note that the set of open balls centered at x with radius 1/n for integers n > 0 form a countable local base at x. An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line). www.all-science-fair-projects.com /science_fair_projects_encyclopedia/First-countable_space   (399 words)

 The Connectivity and Fault-Tolerance of the Internet Topology - Palmer, Siganos, Faloutsos, Faloutsos, Gibbons ... Abstract: In this paper, we apply data mining analysis to study the topology of the Internet, thus creating a new processing framework. To the best of our knowledge, this is one of the first studies that focus on the Internet topology at the router level, i.e., each node is a router. The size (280K nodes) and the nature of the graph are such that new analysis methods have to be employed. citeseer.ist.psu.edu /palmer01connectivity.html   (622 words)

 Topology, continuity and neighbourhood filters   (Site not responding. Last check: 2007-11-07) As I have understood the usual definition of a topological space, X and the topology t ((X,t) being the topological space) are fixed. = = Conversely, given a topology in X, define the neighborhood filters in = the usual way: A(x) consists of all subsets Y of X such that x is in = the interior of Y. Prove that the topology defined as above from this = family of neighborhood filters is the topology you started with. Conversely, given a topology in X, define the neighborhood filters in the usual way: A(x) consists of all subsets Y of X such that x is in the interior of Y. Prove that the topology defined as above from this family of neighborhood filters is the topology you started with. www.thehelparchive.com /new-2370720-279.html   (2914 words)

 Neighbourhood (topology) Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-11-07) Looking For neighbourhood topology - Find neighbourhood topology and more at Lycos Search. Find neighbourhood topology - Your relevant result is a click away! Look for neighbourhood topology - Find neighbourhood topology at one of the best sites the Internet has to offer! www.karr.net /search/encyclopedia/Neighbourhood_%28topology%29   (629 words)

 Evolutionary Algorithms 7 Population models - Parallel implementations A similar strategy to the ring topology is the neighbourhood migration of figure . Like the ring topology, migration is made only between the nearest neighbours. For each subpopulation, the possible immigrants are determined, according to the desired selection method, from the adjacent subpopulations and a final selection is made from this pool of individuals (similar to figure ). www.geatbx.com /ver_3_5/algindex-06.html   (897 words)

 On absolute Lipschitz neighbourhood retracts, mixers, and quasiconvexity by Aarno Hohti   (Site not responding. Last check: 2007-11-07) In this paper we study compact subsets of Euclidean spaces having Lipschitz continuous local mixers. This property, together with the condition of local quasiconvexity, characterizes those subspaces which are Lipschitz neighbourhood retracts. By a suitable compatible re-metrization of a compact space, every (local) mixer of the space can be made Lipschitz continuous. at.yorku.ca /b/a/a/g/01.htm   (102 words)

 Modal Definability in Topology - Gabelaia (ResearchIndex) One of them, namely the notion of compact extension, is a generalization of the concept of Stone- known in topology. 23 The algebra of topology (context) - McKinsey, Tarski - 1944 2 An extension of S4 complete for the neighbourhood semantics.. citeseer.ist.psu.edu /gabelaia01modal.html   (497 words)

 Neighbourhood Urban Sores: On the Interaction Between Segregation, Urban Decay, and Deprived Neighbourhoods (Urban and Regional Planning and Development Series) When Gossips Meet: Women, Family, and Neighbourhood in Early Modern England (Oxford Studies in Social History) Jerusalem 1948: The Arab Neighbourhoods and Their Fate in the War news-server.org /n/ne/neighbourhood.html   (342 words)

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