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Topic: Quotient space

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Quotient

	PlanetMath: quotient space
		As a set, the construction of a quotient space collapses each of the equivalence classes of
		The topology on the quotient space is then chosen to be the strongest topology such that the projection map
		This is version 2 of quotient space, born on 2002-05-23, modified 2003-03-13.
	planetmath.org /encyclopedia/QuotientSpace.html (155 words)

	Quotient space - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.
		The quotient space X/~ is then homeomorphic to Y (with its quotient topology) via the homeomorphism which sends the equivalence class of x to f(x).
		The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
	www.wikipedia.org /wiki/Quotient_space (944 words)

	Encyclopedia: Quotient-space (Site not responding. Last check: 2007-10-19)
		Given a surjective map f : X → Y from a topological space X to a set Y we can define the quotient topology on Y as the finest topology for which f is continuous.
		In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points.
		General topology In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element.
	www.nationmaster.com /encyclopedia/Quotient_space (2234 words)

	Encyclopedia: Quotient space (linear algebra) (Site not responding. Last check: 2007-10-19)
		In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero.
		An immediately corollary (for finite-dimensional spaces) is that the dimension of V is equal to the dimenison of the kernel (the nullity of T) plus the dimension of the image (the rank of T).
		The quotient space is already endowed with a vector space structure by the construction of the previous section.
	www.nationmaster.com /encyclopedia/Quotient-space-%28linear-algebra%29 (505 words)

	Quotient - Wikipedia, the free encyclopedia
		For example, in the problem 6 ÷ 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor.
		A quotient can also mean just the integral part of the result of dividing two integers.
		In more abstract branches of mathematics, the word quotient is often used to describe sets, spaces, or algebraic structures whose elements are the equivalence classes of some equivalence relation on another set, space, or algebraic structure.
	en.wikipedia.org /wiki/Quotient (201 words)

	Quotient space (Site not responding. Last check: 2007-10-19)
		In topology and functional analysis, a quotient space is (intuitively speaking) the result of identifying or "gluing together" certain points of some other space.
		2 Quotient of a vector space by a subspace
		Quotient of a vector space by a subspace
	www.sciencedaily.com /encyclopedia/quotient_space (825 words)

	Quotient space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-19)
		In (The configuration of a communication network) topology and related areas of (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.
		The quotient space X/~ is then (Click link for more info and facts about homeomorphic) homeomorphic to Y (with its quotient topology) via the homeomorphism which sends the equivalence class of x to f(x).
		More generally, suppose X is a space and A is a (A space that is contained within another space) subspace of X.
	www.absoluteastronomy.com /encyclopedia/q/qu/quotient_space.htm (1233 words)

	wiki/Quotient space Definition / wiki/Quotient space Research (Site not responding. Last check: 2007-10-19)
		If a space is connected or path connectedIn 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.
		CompactnessIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is "small" in a certain sense and "contains all its limit points".
		A quotient space of a locally compactIn 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....
	www.elresearch.com /wiki/Quotient_space (2000 words)

	Hausdorff space - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-19)
		space, or separated space, iff, given any distinct points x and y, there are a neighbourhood U of x and a neighbourhood V of y that are disjoint.
		In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring.
		Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers.
	xahlee.org /_p/wiki/Hausdorff_space.html (828 words)

	Quotient space (linear algebra) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-19)
		An immediately corollary (for finite-dimensional spaces) is that the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).
		The quotient space X/M is (Click link for more info and facts about complete) complete with respect to the norm, so it is a Banach space.
		If X is a (A metric space that is linear and complete and (usually) infinite-dimensional) Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.
	www.absoluteastronomy.com /encyclopedia/q/qu/quotient_space_(linear_algebra).htm (650 words)

	Encyclopedia article on Topological space [EncycloZine] (Site not responding. Last check: 2007-10-19)
		A linear graph is a topological space that generalises many of the geometric aspects of graphs with vertices and edges.
		A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
		A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are functionally separated.
	encyclozine.com /Topological_space (2350 words)

	UK Office Space Finder space measurement rules
		This method measures the actual occupiable area of a floor or an office suite and is of prime interest to a tenant in evaluating the space offered by a landlord and in allocating the space required to house personnel and furniture.
		The Rentable Area of an office on the floor shall be computed by multiplying the Usable Area of that office by the quotient of the division of the Rentable Area of the floor by the Usable Area of the floor resulting in the R/U Ratio.
		The Load Factor is the percentage of space on a floor that is not usable, expressed as a percent of Usable Area.
	www.officespacefinder.co.uk /officespacemeasure.html (576 words)

	Hausdorff space Article, Hausdorffspace Information (Site not responding. Last check: 2007-10-19)
		Pseudometric spaces typically arenot Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces.
		In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry,in particular as the Zariski topology on an algebraic variety or the spectrum of a ring.
		Compact preregular spaces are normal, meaning that they satisfy Urysohn'slemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers.
	www.anoca.org /spaces/preregular/hausdorff_space.html (825 words)

	Surface Evolver Documentation - Mathematical model
		The ambient space can be endowed with a general Riemannian metric by putting the keyword METRIC in the datafile followed by the elements of the metric tensor.
		As a generalization of the torus model, you may declare the domain to be the quotient space of R^n with respect to some symmetry group.
		- For a 2 dimensional genus 2 hyperbolic quotient space.
	www.geom.uiuc.edu /software/evolver/html/model.htm (3987 words)

	Station Information - Quotient space
		Consider the set X = R of all real numbers with the ordinary topology, and write x ~ y iff x-y is an integer.
		If Y is some other topological space, then a function f : X/~ → Y is continuous if and only if fop is continuous.
		If g : X → Y is a continuous map with the property that a~b implies g(a)=g(b), then there exists a unique continuous map h : X/~ → Y such that g = hop.
	www.stationinformation.com /encyclopedia/q/qu/quotient_space.html (414 words)

	Algebraic Topology: Topology
		A topological space is a set X together with a collection of subsets OS the members of which are called open, with the property that (i) the union of an arbitrary collection of open sets is open, and (ii) the intersection of a finite collection of open sets is open.
		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.
		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)

	Clearing up the market cycle... best Quotient Vector Space (Site not responding. Last check: 2007-10-19)
		Quotient Vector Space -- from MathWorld Quotient Vector Space -- from MathWorld Suppose that V=\{(x_1,x_2,x_3)\} and W=\{(x_1, 0, 0)\}.
		The purpose of this paper is to draw attention to the widespread occurrence of quotient spaces in.
		In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of "length".
	ascot.pl /th/Fourier5/Quotient-Vector-Space.htm (479 words)

	NSDL Metadata Record -- quotient space
		that satisfies this stronger property is called a quotient map, and given such a quotient map, the space Y is always homeomorphic to the quotient space of X under the equivalence relation...
		As a set, the construction of a quotient space collapses each of the equivalence classes of sim to a single point.
		The topology on the quotient space is then chosen to be the strongest topology such that the projection map pi is continuous.
	nsdl.org /mr/1034456 (182 words)

	Quotient Evolutionary Space: Abstraction of Evolutionary process w.r.t macroscopic propertiesePrints@IISc - Open Access ...
		We map set of all finite populations to a set of macroscopic properties of population those are chosen a prior;; and we call this mapping as evolufionary criteria.
		On the ‘quotient set of populations’ that is induced by evolutionary criteria, we define mathematical structures to define evolutionary change with respect to chosen macroscopic parameters at populational level.
		This allows us to transform the objective defined on the search space that is imposed by the fitness function to an objective on the population space.
	eprints.iisc.ernet.in /archive/00000331 (294 words)

	DPANS94
		Divide ud1 by the number in BASE giving the quotient ud2 and the remainder n.
		Divide d by n3 producing the single-cell remainder n4 and the single-cell quotient n5.
		Note: The requirement to follow the string with a space is obsolescent and is included as a concession to existing programs that use CONVERT.
	maschenwerk.de /HelFORTH/DPANS/dpans6.htm (5969 words)

	Quotient Space articles and news from Start Learning Now (Site not responding. Last check: 2007-10-19)
		We define a topology on the quotient set X/~ (the set consisting of all equivalence classes of ~) as follows: a set of equivalence classes in X/~ is open setopen if and only if their union (set theory)union is open in X.
		Then the quotient topology on X/~ is the finer topologyfinest topology for which q is continuous (topology)continuous.
		- The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
	www.startlearningnow.com /quotient%20space.htm (1051 words)

	iclass.05
		Quotient spaces are rarely considered explicitly in statistical work, but it will be argued in this talk that this oversight is a mistake.
		Quotient spaces do arise naturally in various settings of which the following are typical examples.
		For incomplete data, if the relation between the incomplete response and the complete response is linear, the incomplete response is most naturally viewed as a point in a certain quotient space.
	www.stat.rice.edu /stat/jsm98/session/node182.html (338 words)

	Quotient space : Quotient space (Site not responding. Last check: 2007-10-19)
		Given a surjective map f : X → Y from a topological space X to a set Y we can define the quotient topology on Y as the finest topology for which f is continuous.
		* In general, quotient spaces are ill-behaved with respect to separation axioms.
		* The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
	quotient-space.wikix.ipupdater.com (820 words)

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