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	Discrete space - Wikipedia, the free encyclopedia
		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.
		The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology.
		However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps.
	en.wikipedia.org /wiki/Discrete_space (1058 words)

	PlanetMath: 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.
		This is version 14 of discrete space, born on 2002-02-27, modified 2005-06-19.
	planetmath.org /encyclopedia/Discrete.html (231 words)

	Encyclopedia: Topological space (Site not responding. Last check: 2007-11-06)
		The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety.
		In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties.
		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)

	Base (topology) - Wikipedia, the free encyclopedia
		In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.
		The order topology is usually defined as the topology generated by a collection of open-interval-like sets.
		The metric topology is usually defined as the topology generated by a collection of open balls.
	en.wikipedia.org /wiki/Base_%28topology%29 (793 words)

	Topology glossary - Wikipedia, the free encyclopedia
		The topology T is the smallest topology on X containing B and is said to be generated by B.
		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)

	Discrete space (Site not responding. Last check: 2007-11-06)
		However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by one and nonexpansive maps.
		That is, any function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric space bounded by one is nonexpansive.
		In some sense the opposite of the discrete topology is the trivial topology, which has the least possible number of open sets.
	www.sciencedaily.com /encyclopedia/discrete_space (927 words)

	Continuous function (topology) - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure.
		A continuous functions between two topological spaces stays continuous if we strengthen the topology of the domain space or weaken the topology of the codomain space.
		If a set is given the discrete topology, all functions with that space as a domain are continuous.
	www.wikipedia.org /wiki/Continuous%2B(topology) (849 words)

	PlanetMath: basis (topology)
		A basis for the usual topology of the real line is given by the set of open intervals since every open set can be expressed as a union of open intervals.
		The set of all subsets with one element forms a basis for the discrete topology on any set.
		This is version 11 of basis (topology), born on 2002-01-01, modified 2004-12-20.
	www.planetmath.org /encyclopedia/Base2.html (327 words)

	Discrete and Indiscrete Topologies (Site not responding. Last check: 2007-11-06)
		Given a space t, with its own topology, a set of points s embedded in t is discrete if each point in s is contained in an open set that contains none of the other points of s.
		In the indiscrete topology, only the empty set and the entire set are open and closed.
		In the cofinite topology, all finite sets are closed, along with the entire set (as required).
	www.mathreference.com /top,disc.html (143 words)

	Trivial topology (Site not responding. Last check: 2007-11-06)
		The trivial topology is the topology with least possible number of open sets since definition of a topology requires these two to be open.
		In some sense the opposite of the topology is the discrete topology in which every subset is open.
		The trivial topology belongs to a pseudometric space in which the distance between any points is zero and to a uniform space in which the whole cartesian product X × X is the only entourage.
	www.freeglossary.com /Trivial_topology (777 words)

	Science Fair Projects - Finer topology (Site not responding. Last check: 2007-11-06)
		Any two topologies on X have a meet and join, in the sense of lattice theory; the meet is the intersection, but the join is not in general the union.
		In function spaces and spaces of measures there are often a number of possible topologies.
		See topologies on the set of operators on a Hilbert space for some intricate relationships.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Finer_topology (379 words)

	Comparison of topologies - Enpsychlopedia (Site not responding. Last check: 2007-11-06)
		All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.
		The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.
		In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.
	psychcentral.com /wiki/Coarser_topology (549 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.
		Given a topological space (X,OX) and a function f from X to a set B, we call the topology on B determined by f the quotient topology, and f the corresponding quotient map.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	"Morphing" in Evolutionary Synthesis of Electronic Circuits
		These switches differ from the switches of the discrete-topology version of the method in that instead of being limited to "on" or "off" states, their resistances would be continuously variable between low values (tens to hundreds of ohms, in the "on" state) and high values (~ hundreds of MW in the "off" state).
		By virtue of the intermediate values of the switch resistances, the response of a given circuit topology is almost as if one would combine the responses of several circuit topologies specified by on/off switches.
		The superposition of circuit topologies would be characterized as "fuzzy" because it would blur the borders among distinctive circuit topologies: the resulting circuits would belong, only to certain degrees, to discrete topologies, in each of which any two given components are either connected or not.
	www.nasatech.com /Briefs/Aug02/NPO20837.html (906 words)

	The subspace topology (Site not responding. Last check: 2007-11-06)
		R (with its usual topology/metric) is the discrete topology.
		The subspace topology on the x-axis as a subset of R
		X then the subspace topology is the weakest topology (fewest open sets) on A in which this map is continuous.
	www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L14.html (154 words)

	Exercises 6
		The closure of U in the subspace topology on A is equal to the closure of U in the topology on X.
		Prove that the set of all unbounded open intervals of R forms a sub-basis for the usual topology on R which is not a basis.
		Prove that the discrete topology on R does not have a countable basis.
	www-history.mcs.st-and.ac.uk /~john/MT4522/Tutorials/T6.html (357 words)

	Topology Reading Course
		Students are expected to become familiar with the basic concepts and methodology of point-set topology: separation properties, connectedness, and compactness, as well as subspaces, quotient spaces, and the properties of continuous mappings.
		That is, excluding the trivial smallest topology (consisting of only the empty set and the entire set, called the ``co-discrete'' topology) and the trivial largest topology (which consists of _all_ subsets and is called the ``discrete'' topology), there are 27 nontrivial topologies.
		I mentioned in class that this is trivial if S is finite; in that case, the cofinite topology is the discrete topology.
	www.georgetown.edu /faculty/kainen/topol-02.html (1223 words)

	natural religion > glossary > topological space (Site not responding. Last check: 2007-11-06)
		A topological space is a whole with two parts: A set X, consisting of elements of an arbitrary nature, called points of a given space, and a topological structure or topology, T on this set.
		A refinement of a topology is a cover such that each member of the second lies inside a member of the first, giving higher resolution.
		On a given set, the discrete topology is the finest topology and the indiscrete topology is the coarsest.
	www.naturaltheology.net /Glossary/topoSpace.html (187 words)

	Discrete space (Site not responding. Last check: 2007-11-06)
		Categories more relevant to the metric structure can be found by limiting the morphisms to Lipschitz continuous maps or to nonexpansive maps; however, these categories don't have free objects (on more than one element).
		This homeomorphism is given by ternary notation of numbers.
		In the foundations of mathematics, the study of compactness properties of products of {0,1} is central to the topological approach to the ultrafilter principle, a weak form of choice.
	www.portaljuice.com /discrete_space.html (875 words)

	Discrete Space Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-06)
		Looking For discrete space - Find discrete space and more at Lycos Search.
		Find discrete space - Your relevant result is a click away!
		Look for discrete space - Find discrete space at one of the best sites the Internet has to offer!
	www.stardustmemories.com /encyclopedia/Discrete_space (1203 words)

	Finitism in Geometry
		If space and time are discrete, then the runner, the tortoise, Achilles and all other moving objects simply go through a finite number of space locations in a finite number of time elements and all the problems with supertasks vanish as a one-minute time interval is no longer divisible in a denumerable series of intervals.
		One of the advantages of a discrete approach—and, as a matter of fact, this seems to hold in general for strict finitist proposals—is that definitions that are classically equivalent turn out to be distinct in a strict finitist framework.
		For, in order to determine the dimension, this set must be equipped with a topology and the only possible candidate is the discrete topology.
	plato.stanford.edu /entries/geometry-finitism (4873 words)

	Continuity Topology (Site not responding. Last check: 2007-11-06)
		Last mile access topology must be established to deliver the bandwidth necessary to...
		Continuous functions are fundamental in describing the relationships between topological spaces, and allow simple generalizations of many results from real analysis to be proven.
		If the domain set is given the trivial topology, a topology with only two open sets, and the range set is T
	www.wikiverse.org /continuity-topology (554 words)

	Wood, Zo? Justine (2003-05-16) Computational topology algorithms for discrete 2-manifolds. ... (Site not responding. Last check: 2007-11-06)
		In this thesis, we present novel algorithms guaranteed to identify and isolate handles for various discrete surface representations.
		We also present algorithms to retain or simplify the topology of a reconstructed surface as desired.
		For example, we demonstrate how geometric models can be greatly improved through topology simplification both for models represented by volume data or by triangle meshes.
	resolver.caltech.edu /CaltechETD:etd-05302003-161403 (333 words)

	Digital topology, by Laurence Boxer (Site not responding. Last check: 2007-11-06)
		Although the usual definitions of topology are generally not suited to the analysis of digital pictures, they are easily modified so that notions such as connected component, continuous function, homotopy type, fundamental group, the Hausdorff metric, and others, can be efficiently and profitably employed.
		The mathematical challenge of digital topology lies in the fact that a digital image is a lattice-point approximation of a Euclidean space.
		Since a Euclidean metric imposes a discrete topology on a set of lattice points, it is necessary to use a non-Euclidean foundation as the basis of a theory that allows us to use topological properties in this setting.
	at.yorku.ca /t/a/i/c/06.htm (438 words)

	Discrete group - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-06)
		The Geometry of Discrete Groups (Graduate Texts in Mathematics)
		Actions of Discrete Amenable Groups on Von Neumann Algebras (Lecture Notes in Mathematics)
		Handbook Of Computational Group Theory (Discrete Mathematics and Its Applications)
	encyclopedia.worldsearch.com /discrete_symmetry_group.htm (321 words)

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