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Topic: Neighborhood topology

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Connected space

Jacobian matrix

Cauchy sequence

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	* Topology - (GIS): Definition (Site not responding. Last check: 2007-11-06)
		Topology refers to properties of geometric forms that remain invariant when the forms are deformed or transformed by bending, stretching, and shrinking.
		Topology- Method of determining spatial relationships in vector data models (tells computer what is inside or outside a polygon or which nodes are connected by arcs).
		Topology is the mathematical description and study of continuity--kind of the Platonic version of "the footbone's connected to the anklebone,...
	www.mimihu.com /gis/topology.html (520 words)

	Neighbourhood - Wikipedia, the free encyclopedia
		Neighbourhood is also a term used in mathematics (see the concepts of neighbourhood in topology and the concepts of neighbour and neighbourhood in graph theory) and a song by Space.
		A neighbourhood (CwE) or neighborhood (AmE) 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.
	en.wikipedia.org /wiki/Neighborhood (355 words)

	Topology glossary - Wikipedia, the free encyclopedia
		If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced by T consists of all intersections of open sets in T with A.
		Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
		The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
	en.wikipedia.org /wiki/Topology_Glossary (4512 words)

	PlanetMath: neighborhood
		The meaning of the word neighborhood in topology is not well standardized.
		In fact, almost any argument involving neighborhoods would be unaffected by shrinking a neighborhood to a smaller open set or to an open ball (in the context ot metric spaces).
		This is version 8 of neighborhood, born on 2001-12-28, modified 2004-10-24.
	planetmath.org /encyclopedia/Neighborhood.html (194 words)

	Atlas (topology) - Wikipedia, the free encyclopedia
		In topology, an atlas describes how a complicated space is glued together from simpler pieces.
		For each point in the complicated space, a neighborhood of that point that is homeomorphic to a simple space.
		Each individual map in an atlas of the world gives a neighborhood of each point on the globe that is homeomorphic to the plane.
	en.wikipedia.org /wiki/Atlas_(topology) (412 words)

	cage
		Topologies can be bounded (where they have an edge, and any addresses off that edge are taken as having some background state), or unbounded (usually where they wrap around on themselves, such as for the surface of a sphere).
		Topologies have the ability to normalize addresses, and additionally can make morphologically identical clones of the same size and shape (not necessarily their actual cellular states) for synchronous automata that need to maintain multiple topologies for the sake of simultaneous update.
		Neighborhoods encapsulate the translation of addresses (not cell states) to a list of their neighbors, taking into account the topology they are connected with.
	www.alcyone.com /software/cage (1457 words)

	[No title]
		Topology Glossary Mainly extracted from (a) UC Davis Math:Profile Glossary (http://www.math.ucdavis.edu/profiles/glossary.html) by Greg Kuperberg (http://www.math.ucdavis.edu/profiles/kuperberg.html), and (b) Topology Atlas Glossary (http://www.achilles.net/~mtalbot/TopoGloss.html).
		An early result in topology states that every closed 3-manifold (closed meaning that the manifold is finite and connected but has no boundary) has a Heegaard splitting and a resulting description in terms of a Heegaard diagram, which describes how the two handlebodies are glued together.
		In 3-dimensional topology, a surface in a 3-manifold with the property that no essential circle in the surface bounds a disk in the manifold.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	DMI_AboutNeighborhoods (Site not responding. Last check: 2007-11-06)
		That is, membership in a neighborhood has nothing to do with how close particles are (or get) in the solution space but to how they were arranged before they began hunting for good solutions.
		The third is a Circle Topology in which the particles are arranged in a circle and then each particle includes some (usually small) number of particles on its left and right in its neighborhood.
		The choice of neighborhood topology effects how quickly the influence of a potential best (optimal) solution is propogated to all of the particles.
	www.adaptiveview.com /products/admpso/ospso/javadoc/com/adaptiveview/ospso/dmi/DMI_AboutNeighborhoods.html (583 words)

	Topology
		The second, geometric topology, focuses on the connectivity properties of topological spaces and provides the core results from general topology that serve as background for subsequent courses in geometry and algebraic topology.
		General Topology’s value as a reference work is enhanced by a collection of historical notes for each section, an extensive bibliography, and an index.
		In classical topology, this relation is simple and clear: "An open set is a neighborhood of a point if and only if this point belongs to this open set." In early period of fuzzy topology, "membership relation" was similarly defined.
	www.wordtrade.com /science/mathematics/topology.htm (1869 words)

	PlanetMath: algebraic geometry
		Unfortunately, the only natural definition for ``neighborhood'' on an algebraic object uses the Zariski topology, in which every open set is dense (in an irreducible object).
		Their solution was to introduce the notion of a site, generalizing the notion of a topology.
		Computations in cohomology generally use the same tools as computations in cohomology in algebraic topology: spectral sequences, excision, the Mayer-Vietoris sequence, and so on, with the exception that trivial facts about one-point topological spaces are replaced with difficult algebraic facts (this observation is essentially due to Milne, in his book Étale Cohomology).
	planetmath.org /encyclopedia/AlgebraicGeometry.html (2523 words)

	Topological Preliminaries
		Topology is one of (quite a few) mathematical theories that permeate other branches of Mathematics connecting them into one coherent whole.
		However, as the example of reflection demonstrates, basing our intuitive perception of a topological transformation as an abstraction of a (physical) deformation might be questionable if not misleading.
		Most of the examples will be drawn on the 2-dimensional plane but, given the definitions of the distance and neighborhood could be carried over to the 1- and many dimensional cases.
	www.cut-the-knot.org /do_you_know/topology.shtml (759 words)

	Carfree Cities: City Topology
		The Reference Topology was developed in great detail for Carfree Cities and is available as a highly-detailed poster from Zazzle.com.
		Fewer than about 50 districts indicates either that the topology could be simplified or that the density could be reduced.
		Topologies supporting even larger populations can be imagined.
	www.carfree.com /topology.html (633 words)

	Science Fair Projects - Locally compact space
		In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
		To be precise, a topological space X is locally compact iff every point has a local base of compact neighborhoods.
		All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Locally_compact (1462 words)

	PlanetMath: discrete space
		A space equipped with the discrete topology is called a discrete space.
		The discrete topology is the finest topology one can give to a set.
		The product of an infinite number of discrete spaces is discrete under the box topology, but if an infinite number of the spaces have more than one element, it is not discrete under the product topology.
	planetmath.org /encyclopedia/Discrete.html (231 words)

	Topology, continuity and neighbourhood filters (Site not responding. Last check: 2007-11-06)
		Given a family of neighborhood filters, define the collection of open sets to consists exactly of all sets U such that U in A(x) for each x in X. Prove that this set is a topology.
		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.
		If you have a neighborhood filter for each point of X, you define the topology on X by U subset of X is open if and only if for all x in U, U in A(x).
	www.thehelparchive.com /new-2370720-279.html (2898 words)

	In The Neighborhood (Site not responding. Last check: 2007-11-06)
		It's customary to treat the relativistic spacetime manifold as an ordinary topological space with the same topology as a four-dimensional Euclidean manifold, denoted by R
		"...the neighborhood of a given point is the set of all points such that their coordinates differ only a little from those of the given point."
		Interestingly, it is often suggested that the usual Euclidean topology of spacetime might break down on some sufficiently small scale, such as over distances on the order of the Planck length of roughly 10
	www.mathpages.com /rr/s9-01/9-01.htm (784 words)

	UTOPIA Technology: Network Topology (Site not responding. Last check: 2007-11-06)
		A ring topology provides redundant routes for network traffic: if fiber is cut anywhere along the ring, traffic can reverse itself and travel the opposite direction to reach its destination.
		Once the fiber reaches a neighborhood, a star topology delivers a dedicated strand of fiber to the connected residence or business.
		The physical design of the network is supported by its logical design, which effectively divides the network into functional layers.
	www.utopianet.org /technology/topology.htm (131 words)

	ONT Re: Topology (Site not responding. Last check: 2007-11-06)
		every set to which a point belongs is a neighborhood of it.
		open iff it is a neighborhood of each of its points.
		The 'neighborhood system' of a point is the family
	0-suo.ieee.org.csulib.ctstateu.edu /ontology/msg03869.html (338 words)

	[No title]
		Dynamic contextual topologies Of course, the components of a book are frozen into a single context by the order in which they are presented in relation to each other.
		Problems with these contextual topologies The contextual topologies that were discussed above are extreme in terms of their relationship between context and content.
		Contextual navigation by "neighborhood switching" is one of the characteristics of a concept browser, which (to the author's knowledge) distinguishes it from any other knowledge management tool that is available today.
	kmr.nada.kth.se /papers/ConceptualBrowsing/ConceptBrowser.doc (4467 words)

	Topology MAT 530
		Some other examples of topological spaces: the 3 essentially different topologies on a 2-point set, the order topology of a linearly ordered set, we also know how to define topology on a partially ordered set such that any pair of elements admits a lower bound.
		This is the largest (finest, strongest) topology such that the canonical projection (from the space to the quotient-space) is continuous.
		A counterexample is the set of all rational numbers with the topology induced from the reals (which is the same as the order topology) --- all rationals are separate connected components, but they are not open.
	www.math.sunysb.edu /~timorin/mat530.html (2904 words)

	[Nest-challenge] Group Management in NEST
		Neighborhood Maitenance Component: ------------------------------------- The main call expored is: command GetNeighborhoodInfo() It returns a data structure with information regarding neighborhood health.
		notice that it is not necessarily immediate neighborhood but possibly extended to 2-3 hops from a node.
		possibly, but better if neighborhood is maintained as a separate component: - neighborhood maintenance can be done locally and independently of routing (using messages from other components or their own).
	mail.millennium.berkeley.edu /pipermail/nest-challenge/2002-October/000025.html (839 words)

	Module Documentation - AI::NeuralNet::SOM 0.02 (Site not responding. Last check: 2007-11-06)
		Since this mapping is assumed to be continuous along some hypothetical "elastic surface", it may be self-evident how the unknown data are interpreted by means of interpolation and extrapolation with respect to these calibrated points.
		Variable $topology may be either "rect" or "hexa", $neighborhood may be "bubble" or "gaussian".
		The format of the entries is similar to that used in the input data files, except that the optional iitems on the first line of data files (topology type, x- and y-dimensions and neighborhood type) are now compulsory.
	aspn.activestate.com /ASPN/CodeDoc/AI-NeuralNet-SOM/SOM.html (1842 words)

	[No title]
		Do not fiddle with it.") (defcustom mtorus-current-topology nil "Current topology to use when navigating through the mtorus universe." :group 'mtorus-topology) ;; (defcustom mtorus-topology-alist ;; '((neighborhood) ;; (neighborhood-selectors)) ;; "Alist of function specifiers and corresponding funs used ;; for determining topology issues in the `mtorus-universe'.
		NAME is the name of the topology and PROPERTIES is a list of property names as keywords that describe the topology in detail.
		A topology for mtorus is a function which takes an mtorus-element as argument and returns a `neighborhood', i.e.
	www.math.tu-berlin.de /~freundt/mtorus/mtorus-topology.el (790 words)

	Experiments in Topology (Site not responding. Last check: 2007-11-06)
		Network topology is considered in chapter 8, with the famous Koenigsberg bridge problem leading off the discussion.
		The author does not want to leave the reader with the impression that topology is all scissors, paper, and tape, so he devotes the last two chapters of the book to point-set topology.
		It is quite difficult to explain to beginning readers and students of topology what a neighborhood actually does without having the notion of a metric or distance, but the author does a fairly good job here.
	www.textkit.com /0_0486259331.html (858 words)

	AdaptiveView.com - An Introduction to Particle Swarm Optimization (Site not responding. Last check: 2007-11-06)
		During that time, members of the still unaffected neighborhoods continue to hunt in their area of the solution space following the scent of what they think is best.
		It is this personal best in combination with its neighborhood's best that influences where the particle moves to through the solution space (i.e., to how its velocity and direction along each dimension of the solution space will be altered each time it moves).
		It connects particles and the behavior of particles through its implementation of neighborhoods and the inclusion of the neighborhood-best fitness in the factors that determine how a particle will move.
	www.adaptiveview.com /articles/ipsoprnt.html (2082 words)

	Topology
		Excellent, well-written introductory text covers the algebra of subsets and of rings and fields of sets, complementation and ideal theory in the distributive lattice, closure function, neighborhood topology, topological maps, the derived set in T1-space and the topological product, much more.
		Fresh approach explains nontrivial applications of metric space topology to analysis; topics from elementary algebraic topology focus on concrete results with minimal formalism.
		Comprehensive coverage of elementary general topology as well as algebraic topology, specifically 2-manifolds, covering spaces and fundamental groups.
	www.doverdirect.com /0486656764.html (297 words)

	Abstract (Site not responding. Last check: 2007-11-06)
		The mobility of sensor nodes could lead to network topologies wherein accurate computation of absolute position of all the sensor nodes may not be possible.
		In this paper, we propose a topology based localization approach that suggests a best possible approximate position for sensor nodes for which computation of exact absolute position is not possible.
		Each sensor node strives to improve its localization by constantly monitoring its neighborhood and requesting an associated c-node to recompute position whenever neighborhood topology changes.
	dit.unitn.it /wons/Abstracts/abstract-0015.html (289 words)

	Re: neighborhood -topology~~~
		A function can be continuous at a point without being continuous.
		If f is not continuous then the f inverse image of an (open) neightborhood won't always be an (open) neighborhood.
		Since "continuous at" does not require continuity it needs to be defined in a way that does not rely on continuity, i.e., without expecting the inverse image of an open set to be open.
	www.usenet.com /newsgroups/sci.math/msg21053.html (173 words)

	Neighborhood Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-06)
		Looking For neighborhood - Find neighborhood and more at Lycos Search.
		Find neighborhood - Your relevant result is a click away!
		Neighbourhood is also a term used in mathematics (see the concepts of neighborhood in topology and the concepts of neighbor and neighborhood in graph theory) and a song by Space.
	www.variedtastes.com /encyclopedia/Neighborhood (459 words)

	KLAT2's Flat Neighborhood Network
		This may seem like a trivial concern, but flat neighborhood designs do not necessarily have good wiring locality properties and, in the general case, are not regular (i.e., often have no symmetry).
		A number of generations after finding a solution to the simplified network design problem, the population of network designs is scaled back to the original problem size, and the GA resumes with the designer-specified evaluation function.
		Nine of these switches form the flat neighborhood network's switching fabric; the tenth is used exclusively for (1) I/O to other clusters, (2) multicast, and (3) connection of the two ``hot spare'' Athlon PCs.
	www.linuxshowcase.org /2000/2000papers/papers/dietz/dietz_html (5126 words)

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