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NationMaster - Encyclopedia: Quotient space In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by collapsing N to zero. 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). 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   (2419 words)

PlanetMath: quotient space is the topology whose open sets are the subsets 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/QuotientTopology.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. 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 if and only if their union is open in X. 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). en.wikipedia.org /wiki/Quotient_space   (965 words)

Diffeology A diffeological space has the D-topology: the finest topology such that all plots are continuous. If X is a diffeological space and ~ is some equivalence relation on X, then the quotient set X/~ has the diffeology generated by all compositions of plots of X with the projection from X to X/~. Note that the quotient D-topology[?] is the D-topology of the quotient diffeology. www.ebroadcast.com.au /lookup/encyclopedia/di/Diffeology.html   (502 words)

General Topology - NoiseFactory Science Archives (http://noisefactory.co.uk) This is indeed a topology, called the quotient topology induced on Y by If we assign P the discrete topology, in which every subset is open, these will include all the inverse images of open sets in the various factor spaces. The standard topologies on N, Z, Q, and R are all (defined to be) their order topologies. noisefactory.co.uk /maths/topology.html   (4788 words)

Topological space - Wikipedia, the free encyclopedia The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X  →  Y is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. en.wikipedia.org /wiki/Topological_space   (1934 words)

Quotient Space   (Site not responding. Last check: 2007-10-17) Let q, the quotient space (or factor space) of s, be the equivalence classes of s, as determined by r. In general, the quotient of an n dimensional ball by its boundary gives an n dimensional sphere. Therefore the quotient space is the compactification of the open ball, where p is the point at infinity. www.mathreference.com /top,qspc.html   (470 words)

[No title]   (Site not responding. Last check: 2007-10-17) For example the discrete >>> topology, that (is a metric topology, but) is not a normizable >>> topology. I mean a >topology that makes a finite-dimensional linear space V into a >topological linear space and that is not the euclidean topology (up to >a homeomorphism). Conversely, if E is any real or complex vectorspace, and K is any subspace such that E/K is finite-dimensional, then E can be given the weakest topology (fewest open sets) such that K is closed and the quotient topology on E/K is linearly homeomorphic to the Euclidian topology. www.math.niu.edu /~rusin/known-math/01_incoming/TVS   (297 words)

Quotient topology if Y has the smallest possible topology, with only the empty set and Y open, the map is always continuous. in general quotient constructions are techniques for constructing "maximal" targets for surjective maps subject to a certain condition. then try to convince yourself that the quotient topology is exactly the usual topology on the circle. www.physicsforums.com /showthread.php?t=54155   (549 words)

Topology MAT 530   (Site not responding. Last check: 2007-10-17) 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 /~azinger/mat530/fall04/index.htm   (2907 words)

Springer Online Reference Works   (Site not responding. Last check: 2007-10-17) on which two structures are given — a group structure and a topology, such that the group operations are continuous. of cosets is given the quotient topology with respect to the canonical mapping from The topology arising from the uniform structure is the same as the original topology on the group. eom.springer.de /T/t093070.htm   (1330 words)

Topology I It covers open and closed sets, continuity, compactness, connectedness, and quotient spaces. Be able to define and use the concepts of continuity, compactness, and connectedness. Understand and be able to construct spaces using the quotient topology. www.calpoly.edu /~math/gcourses/math540.html   (76 words)

Math 335 Course Description In addition to a thorough coverage of point set topology, to which students are introduced in Math 317, it provides an introduction to algebraic topology. The fundamental group and covering spaces are discussed in depth to prepare students for either Topology II at Haverford or a graduate course in Topology at Bryn Mawr. In recent years, Topology II at Haverford has covered simplicial homology theory (since graduate programs teach the less concrete singular homology theory), but if students are interested it could cover differential geometry instead. www.haverford.edu /math/courses/math335.html   (237 words)

Amazon.com: Topology (2nd Edition): Books: James Munkres   (Site not responding. Last check: 2007-10-17) I must say I was quite confused when I began the actual topology portion of the book (chapter 2), but this was due to the difficulty of topology, in general, and not the book itself. If you are searching for an introduction to point-set topology that will give you a solid grounding in the basics of point-set topology, but at the same time will give it to you in an easily approached manner, than this book is for you. Certain topics, such as the quotient topology and quotient maps, come out of nowhere and are presented in a muddled fashion. www.amazon.com /Topology-2nd-James-Munkres/dp/0131816292   (1648 words)

AdjunctionSpace.htm is defined as the finest topology on X for which all injections which is onto, we define a topology on We can also consider the quotient topology to be the "largest" topology (most open sets) for which www.umsl.edu /~siegel/Topology/AdjunctionSpace.htm   (105 words)

Elliptic Curves and Modular Functions If the set which is acted upon has a topology, then the set of orbits also has a topology which is called the "quotient topology", and the resulting topological space is called the "quotient space". Since a Riemann surface has a topology, a group of analytic transformations acting upon it defines a quotient space, which is also a Riemann surface. If the surface happens to be a quotient space with respect to a symmetry group on another surface, then the space of all its meromorphic functions corresponds to a very special class of functions on the "larger" surface: the automorphic functions. www.mbay.net /~cgd/flt/flt05.htm   (2994 words)

200 Level Courses This course is not a "continuation" of 22M:132: it is not a course in general topology. You may have learned enough topology ideas from another prior course (like our Topology 22M:130 or old 22M:115) to be well prepared. Special note for students thinking of taking their Ph.D. Comprehensive Exam in Topology: Department rules allow students to take either a written exam based on 22M:201-200 type material, or exams (oral or written) designed individually for students with more clear research plans. www.math.uiowa.edu /grad/gradcourses200fa2004.html   (526 words)

The identification topology a circle and it would be a good idea to consider it with its subspace topology inherited from the plane. Then every equivalence class has a unique representative, but one should arrange things so that classes with representatives close to 1 should be near the one with representative 0. It is easy to see that this is the same as the subspace topology on the circle as a subset off the plane. www-groups.dcs.st-and.ac.uk /~john/MT4522/Lectures/L16.html   (534 words)

Tulane Math Graduate qualifying exam syllabi General definitions of topological spaces: open and closed sets, bases and subbases for topology, subspaces and subspace topology, continuous functions, homeomorphism. Metric spaces: bases and subbases for topology, equivalent metrics, Cauchy sequences and completeness, Baire category Theorem (SSA). Quotient spaces and quotient topology: definitions and use in examples such as showing a map from a quotient space is continuous, or there is a homeomorphism from the quotient www.math.tulane.edu /graduate/qualifying/topology.html   (673 words)

UM Department of Mathematics: Graduate Description and in-depth study of the basic algebraic structures: groups, rings, fields including: set theory, relations, quotient groups, permutation groups, Sylow's Theorem, quotient rings, field of fractions, extension fields, roots of polynomials, straight-edge and compass solutions, and other topics. Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differential manifolds, tangent spaces, vector fields, submanifolds, inverse f unction theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces. Further topics in algebraic topology typically taken from: obstruction theory, cohomology operations, homotopy theory, spectral sequences and computations, cohomology of groups, characteristic classes. www.math.lsa.umich.edu /graduate/byarea.shtml   (3027 words)

Topology Problem Sets   (Site not responding. Last check: 2007-10-17) be the identification space resulting from the quotient topology. A topological space X is said to be locally compact if each point x in X has a compact set N containing an open set U containing x for each x [thus x in open U in compact N]. Create a topology on Y by defining the collection of open sets in Y to be all sets of the following types: www.geneseo.edu /~johannes/338PS1.html   (362 words)

pushnote.htm That it is a topology, again, follows quickly from the Boolean Algebra above. It is a simple argument to check that is a homeomorphism. as having the push out topology, the glueing lemma is just a restatement of the definition of the push out topology. www.umsl.edu /~siegel/Topology/pushnote.htm   (96 words)

Subspaces and Quotients for LCS is given the quotient topology, then this is exactly the topology induced by The first statement is left as an exercise. For the forward direction, by the remark for a quotient topology on an LCS, www.math.unl.edu /~s-bbockel1/929/node7.html   (91 words)

quotient - OneLook Dictionary Search Tip: Click on the first link on a line below to go directly to a page where "quotient" is defined. Quotient (m), quotient, quotient (m) : AllWords.com Multi-Lingual Dictionary [home, info] Phrases that include quotient: intelligence quotient, respiratory quotient, quotient verdict, quotient ring, quotient space, more... www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=quotient   (278 words)

Maths - icecube’s keep The project is in General Topology and we have to give a whole discussion about Quotient There are quite a few different ways of viewing quotient mappings; looking at them as being identification maps (the standard “glueing” explanation) is the most geometrical, but that doesn’t explain explicitly what the open sets are. Also, the description of the quotient topology as a universal property is definitely worth thinking about and trying to figure out what it means, because the description of things as universal properties becomes more and more common as you go up the strata in maths. www.maths.tcd.ie /~icecube/maths   (712 words)

Mathematics Topology Homework Help You are reading an article that reported the results of a study that examined the relationship between duration of television watching f...continues General and Differential Topology in Graduate School level This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. www.brainmass.com /homeworkhelp/math/topology/pg4   (398 words)

Northeastern University, Department of Mathematics The Topology exam covers the material taught in MTH G121 (Topology 1), and the beginning of MTH G221 (Topology 2). James R. Munkres, Topology, 2nd Edition, Prentice Hall, 2000. Glen Bredon, Topology and Geometry, Springer-Verlag, GTM #139, 1997. www.math.neu.edu /grad/quals/top-syllabus.html   (127 words)

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