
	Where results make sense

Topic: Quotient map

Related Topics

Open map

Open set

ManiFold

In the News (Sun 20 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Quotient space - Wikipedia, the free encyclopedia
		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.
		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 continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image).
	en.wikipedia.org /wiki/Quotient_space (944 words)

	Quotient space (Site not responding. Last check: 2007-11-03)
		In topology and functional analysis, a quotient space is (intuitively speaking) the result ofidentifying or "gluing together" certain points of some other space.
		The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined onX that respect the equivalence relation (in the sense that they send equivalent elements to the same image).
		The quotient space is already endowed with a vectorspace structure by the construction of the previous section.
	www.therfcc.org /quotient-space-214765.html (725 words)

	Quotient Space [Definition] (Site not responding. Last check: 2007-11-03)
		We define a topology on the quotient setIn mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
		The quotient space X/~ together with the quotient map q : X â’ X/~ is characterized by the following universal propertyIn category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions.
		quotient groupIn 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.wikimirror.com /Quotient_space (2320 words)

	Complexes of Modules (Site not responding. Last check: 2007-11-03)
		The homology of the complex in degree n is the quotient of the kernel of the n^(th) boundary map by the cokernel of the boundary map of degree n - 1.
		The maps in the complex are all multiplied by the scalar (- 1)^n.
		The splice of the complex C with the complex D along the map f from the last term of C to the first term of D. The degree of the last term of the splice is the same as the degree of the last term of the complex D. Extensions
	magma.maths.usyd.edu.au /magma/htmlhelp/text797.htm (1807 words)

	Hausdorff space - Wikipedia, the free encyclopedia
		Pseudometric spaces typically are not 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.
		Let f : X → Y be a quotient map with X a compact Hausdorff space.
	en.wikipedia.org /wiki/Hausdorff_space (1227 words)

	Jacobson Semisimple and Quotient Rings
		The quotient map induces a group homomorphism from the units of r onto the units of r/j.
		The quotient map r/j induces an r module homomorphism from the cosets of h in r to the cosets of y in r/j.
		Verify that this is indeed a homomorphism, that the kernel is precisely h (the 0 coset), and that every coset of y is accessible, hence the homomorphism is an isomorphism.
	www.mathreference.com /ring-jr,ss.html (734 words)

	[No title]
		The map is a delooping of the Adams opera* tion map* _____________ 1991 Mathematics Subject Classification.
		An important special case arises when M is the mapping cone CN = C1N of the * identity map* on a finitely generated projective R-module N, so that M is an acyclic chai* *n complex of length 1, with a copy of N in degrees 0 and 1.
		By multilinearity (2.1) the quotient Wp=Wp-1 is N1.
	hopf.math.purdue.edu /Grayson/adamsops.txt (3544 words)

	Adjunction space - Wikipedia, the free encyclopedia
		Let f : A → X be continuous map (called the attaching map).
		The topology, however, is specified by the quotient construction.
		are the maps obtained by composing the quotient map with the canonical injections into the disjoint union of X and Y.
	en.wikipedia.org /wiki/Adjunction_space (374 words)

	Topology glossary - Wikipedia, the free encyclopedia
		Two continuous maps f, g : X → Y are homotopic (in Y) if there is a continuous map H : X × [0, 1] → Y such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X.
		A partition of unity of a space X is a set of continuous functions from X to [0, 1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.
		A path in a space X is a continuous map f from the closed unit interval [0, 1] into X.
	en.wikipedia.org /wiki/Topology_Glossary (4500 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Such a map is called a {\it special generic map}, which was first defined by Burlet and de Rham for $(n, p) = (3, 2)$ and later extended to general $(n, p)$ by Porto, Furuya, Sakuma and Saeki.
		In this paper, we study the global topology of such maps for $p = 3$ and give various new results, among which are a splitting theorem for manifolds admitting special generic maps into ${\bf R}^3$ and a classification theorem of 4- and 5-dimensional manifolds with free fundamental groups admitting special generic maps into ${\bf R}^3$.
		Throughout the paper, manifolds and maps are of class $C^\infty$ and the symbol ^^ ^^ $\cong$" denotes a diffeomorphism between manifolds or an appropriate isomorphism between algebraic objects.
	home.imf.au.dk /esn/preprints/090 (7797 words)

	PlanetMath: quotient space (Site not responding. Last check: 2007-11-03)
		to its equivalence class is always a continuous map.
		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.
	www.planetmath.org /encyclopedia/QuotientSpace.html (155 words)

	Reference re Prov. Electoral Boundaries (Sask.), [1991] 2 S.C.R. 158 (Site not responding. Last check: 2007-11-03)
		The resulting distribution map, unlike the one it replaced, revealed a number of ridings with variations in excess of 15 percent from the provincial quotient and indicated a problem of under-representation in urban areas.
		A glance at the 1981 map demonstrates that each and every southern riding, whether urban or rural, is within 15 percent of the provincial quotient -- the figure obtained by dividing the southern voting population by the number of southern ridings.
		For instance, the 1981 map provides proof that it is possible in Saskatchewan to achieve equality within 15 percent of the provincial quotient for all southern constituencies while still addressing other relevant considerations such as the differing nature of rural and urban interests.
	www.lexum.umontreal.ca /csc-scc/en/pub/1991/vol2/html/1991scr2_0158.html (11949 words)

	4Reference \|\| Quotient space (Site not responding. Last check: 2007-11-03)
		In topology, a quotient space is (intuitively speaking) the result of identifying or "gluing together" certain points of some other space.
		The map p is continuous; in fact, the topology on X/~ is the finest (the one with the most open sets) which makes p 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.4reference.net /encyclopedias/wikipedia/Quotient_space.html (447 words)

	On The Pullback Stability Of A Quotient Map With Respect To A Closure Operator (ResearchIndex)
		There are well-known characterizations of the hereditary quotient maps in the category of topological spaces, (that is, of quotient maps stable under pullback along embeddings), as well as of universal quotient maps (that is, of quotient maps stable under pullback).
		These are precisely the so-called pseudo-open maps, as shown by Arhangel'skii, and the bi-quotient maps of Michael, as shown by Day and Kelly, respectively.
		In this paper hereditary and stable quotient maps are characterized...
	citeseer.ist.psu.edu /452064.html (421 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Let p be the quotient map from X onto Q. Then: p is an open map; i.e., whenever G is an open subset of X, then p(G) is an open subset of Q. Also: If K is closed, then Q is Hausdorff.
		The map f factors naturally through p -- we have a map g from Q to Y defined by g(p(x)) = f(x).
		Verify that g is a continuous linear bijection from Q onto Y; hence it is an isomorphism of topologies; hence it is an open map.
	math.vanderbilt.edu /~schectex/ccc/addenda/openmap.html (318 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		A Riemann surface is a topological space with a collection of "charts": homeomorphisms phi_i of open subsets of X to open subsets of the complex plane, such that the charts cover X, and wherever two charts overlap, the functions phi_i composed with {phi_j}^{-1} is a holomorphic function.
		By the Riemann mapping theorem, though, there is an isomorphism between any two bounded simply connected open subsets of the complex plane; there's a 1-1 map from one to the other which has an inverse which is also holomorphic.
		Rational functions are holomorphic maps from the sphere to itself; the points where the denominator goes to zero but the numerator doesn't get mapped to infinity, and infinity gets mapped to the limit of the function as you approach infinity.
	www.math.niu.edu /~rusin/papers/known-math/95/modularity (2492 words)

	The Variety Generated by Banach Spaces (Site not responding. Last check: 2007-11-03)
		Every Hausdorff topological group is a quotient space of a topological space which admits a continuous metric.
		Then by Proposition 3.1, G is a quotient space of a topological space Xwhich admits a continuous metric.
		Further, by Proposition 3.2, G is a quotient topological group of FA(X), the free abelian topological group on X.
	cedir.uow.edu.au /Projects/math_test/node3.html (755 words)

	PlanetMath:
		quotient ideal (=quotient of ideals) owned by pahio
		quotient map (in quotient space) owned by djao
		quotient topology (in quotient space) owned by djao
	planetmath.org /encyclopedia/Q (165 words)

	Abelian, Nilpotent and Soluble Quotient
		The functions returning the abelian quotient or the abelian quotient invariants report an error if the abelian quotient cannot be computed, for example, because the relation matrix is too large.
		A nilpotent quotient algorithm constructs, from a finite presentation of a group, a polycyclic presentation of a nilpotent quotient of the finitely presented group.
		G has infinite nilpotent quotients if and only if G/G_1 (the maximal abelian quotient of G) is infinite and a prime p divides a finite cyclic factor of a nilpotent quotient if and only if p divides a cyclic factor of G/G_1.
	www.umich.edu /~gpcc/scs/magma/text297.htm (2923 words)

	Algebraic Topology: Topology
		A map or continuous function from a topological space (X,OX) to a topological space (Y,OY) is a function from X to Y such that the preimage of any member of OY is a member of OX.
		A homeomorphism is a bijective map of which the inverse is a map, too.
		Also, there need not be any non-constant maps from [0,1] to a connected topological space (indeed, there exist countable connected Hausdorff spaces), and in such a case all path-components are singletons.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	Tensoring with a Quotient Ring
		Verify that this is a bilinear map, and if r is commutative it respects the action of r.
		Let t be the cosets of km in jm, and map x in j and y in m to xy, a representative of km in jm.
		Use Bezout's identity to show that the sum of two ideals is spanned by the gcd of their generators.
	www.mathreference.com /mod-pit,tqr.html (1297 words)

	Quotient Groups
		The generators of the quotient are images of the generators of G. G / N : GrpPerm, GrpPerm -> GrpPerm
		Given a normal subgroup N of the permutation group G, construct the quotient of G by N. Currently, the quotient group is constructed in its regular representation, so that the application of this operator is restricted to the case where the index of N in G is a few thousand.
		This function returns the class c nilpotent quotient of G, together with the epimorphism pi from G onto this quotient.
	www.enseignement.polytechnique.fr /profs/informatique/Eric.Schost/X2001/Maj1/htmlhelp/text315.htm (513 words)

	[No title]
		The quotient maps bAji Hnjinduce maps from bAjto the inverse limit, and one can see that the kernel of the map to the limit is zero.
		Let R be the inverse limit of the RT,S under the apparent maps (map each polynomial generator to the generator of the same name if present, and to zero otherwise).
		Since æ is the inverse limit of the maps æn, and since each æn is a restrict* ion 0 map*, æ commutes with the action of fSq.
	hopf.math.purdue.edu /Palmieri/quotient.txt (3601 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Since some people don't know how to take the quotient of a ring R by an ideal I, give them the example of the ring of integers and the ideal generated by an integer n.
		Let p=5 and find elements in the valuation ring of the 5-adic absolute value, the valuation ideal, and find an integer not equal to 1 which maps to 1 under the quotient map.
		Write down all the elements in the valuation ring which map to 1 under the quotient map.
	www-math.mit.edu /~coneil/704/lecture4.html (152 words)

	Subgroups, Quotient Groups and Extensions
		Construct the subgroup N of the pc-group G as the normal closure of the subgroup generated by the elements specified by the terms of the generator list L. The possible forms of a term L[i] of the generator list are the same as for the sub-constructor.
		Construct the quotient Q of the pc-group G by the normal subgroup N, where N is the smallest normal subgroup of G containing the elements specified by the terms of the generator list L. The possible forms of a term L[i] of the generator list are the same as for the sub-constructor.
		The extension K will have a normal subgroup G isomorphic to G, while the quotient group K/G is isomorphic to H. The homomorphism phi is given by the sequence of maps f.
	www.math.uiuc.edu /Software/magma/text219.html (1152 words)

	[No title]
		The Exposure Quotient is obtained by measuring Electric Field strengths and referencing the results to the relevant ICNIRP maximum public exposure levels.
		The Exposure Quotient is expressed as a scientific number which can be interpreted as a fraction of the maximum exposure level.
		The total exposure quotient for the original band (30MHz — 2GHz) is 1.82e-3 These results show that most of the energy is transmitted in the 30 — 860MHz section of the band.
	www.ofcom.org.uk /static/archive/ra/topics/mpsafety/school-audit/multiuser-reports/sandyheath-comparison.doc (2469 words)

Try your search on: Qwika (all wikis)