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Topic: Topological

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	Topological group - Wikipedia, the free encyclopedia
		In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps.
		An example of a topological group which is not a Lie group is given by the rational numbers Q with the topology inherited from R.
		In many ways, the locally compact topological groups serve as a generalization of countable groups, while the compact topological groups can be seen as a generalization of finite groups.
	en.wikipedia.org /wiki/Topological_group (813 words)

	Topology - Wikipedia, the free encyclopedia
		Topology is concerned with the study of the so-called topological properties of figures, that is to say, properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms).
		The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
		In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
	en.wikipedia.org /wiki/Topology (1344 words)

	Topological vector space - Wikipedia, the free encyclopedia
		As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space.
		The elements of topological vector spaces are typically functions, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.
		A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by −1).
	www.wikipedia.org /wiki/Topological_vector_space (1006 words)

	Topological space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.
		A linear graph is a topological space that generalises many of the geometric aspects of (A drawing illustrating the relations between certain quantities plotted with reference to a set of axes) graphs with vertices and edges.
		Topological spaces can be broadly classified according to their degree of connectedness, their size, their degree of compactness and the degree of separation of their points and subsets.
	www.absoluteastronomy.com /encyclopedia/t/to/topological_space.htm (3343 words)

	Directory - Science: Physics: Quantum Mechanics: Quantum Field Theory: Topological (Site not responding. Last check: 2007-11-07)
		Topological Quantum Field Theories · Topological quantum field theories can be used as a powerful tool to probe geometry and topology in low dimensions.
		Geometry of 2D Topological Field Theory · These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories.
		Topological Quantum Field Theory, a Progress Report · A brief introduction to Topological Quantum Field Theory as well as a description of recent progress made in the field is presented.
	www.incywincy.com /default?p=210316 (465 words)

	Dynamic Topological Logic (Site not responding. Last check: 2007-11-07)
		Topological dynamics studies the asymptotic properties of continuous maps on topological spaces.
		Let a dynamic topological system be a topological space X together with a continuous function f.
		Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality □ (interior), and two temporal modalities, ○ (a circle for "next") and * (an asterisk for "henceforth"), both interpreted using the continuous function f.
	individual.utoronto.ca /philipkremer/DynamicTopologicalLogic.html (577 words)

	Math 121 Course Information (Site not responding. Last check: 2007-11-07)
		Topologically speaking, each is a sphere with one handle, and each can be continuously deformed to the other.
		One method for studying topological spaces is to assign algebraic objects, such as groups or vector spaces, to a topological space.
		Topological spaces are introduced, along with the separation axioms and various notions as compactness, local compactness, connectedness, and path connectedness.
	www.math.ucla.edu /undergrad/courses/math121 (411 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.
		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.
		A topological space is said to be locally P for some property P when for each point x and neighbourhood U of x there is a set A contained in U and containing a neighbourhood of x that has property P.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	RMK articles : Topological Torsion and Spin in Plasma Reconnection Processes
		When the second Poincare invariant vanishes, the closed integrals of Topological Torsion are deformation invariants (hence, topological properties) for any evolutionary process described by a singly parameterized vector field.
		Topological Spin generates a topological property of the fields which is distinct from the properties of Topological Torsion.
		The four divergence of the Topological Spin is proportional to the first Poincare invariant, and is equal to the Lagrangian of the field including interactions between the currents and the potentials: ((B.H) - (D.E)) - ((A.J) - rho.phi).
	www22.pair.com /csdc/pd2/pd2fre39.htm (459 words)

	Topological M-Theory \| Musings
		Topological String Theory is a rich and beautiful subject.
		One of the “axioms” of topological M-Theory is that, on
		I was thinking that “topological” meant something stronger than “holomorphic” — that it meant independence of (some of) the moduli.
	golem.ph.utexas.edu /~distler/blog/archives/000489.html (886 words)

	Topological Spaces: From Distance to Neighborhood by Buskes and van Rooij - Apronus.com
		Elementary Topology preeminently is a subject with an extensive array of technical terms indicating properties of topological spaces.
		To meet them halfway, in Chapter 18 we briefly introduce and discuss a number of topological properties, but even there we do not touch on paracompactness, complete normality, and extremal disconnectedness - just to mention three terms that are not really esoteric.
		As, however, a rigorous theory of topological spaces must have a firm base in Analysis, we start with a brief axiomatic treatment of the real-number system, explaining what axioms are and what purpose they serve.
	www.apronus.com /math/buskes_vanrooij_topsp.htm (780 words)

	Topological condensate concept
		One ends up with the concept of the topological condensate from the requirement that both TGD as a Poincare invariant gravity and TGD as generalization of the string model pictures make sense as appropriate limiting cases.
		Topological condensate has a hierarchical structure and one can associate a characteristic length scale spesifying the minimum size of the 3-surfaces on each level of the hierarchy.
		This picture of topological condensate is consistent with the general properties of the Kähler action.
	www.physics.helsinki.fi /~matpitka/topcond.html (487 words)

	Topological Sorting (Site not responding. Last check: 2007-11-07)
		Topological sorting can be used to schedule tasks under precedence constraints.
		These precedence constraints form a directed acyclic graph, and any topological sort (also known as a linear extension) defines an order to do these tasks such that each is performed only after all of its constraints are satisfied.
		Since the basic topological sorting algorithm will get stuck as soon as it identifies a vertex on a directed cycle, we can delete the offending edge or vertex and continue.
	www.cs.toronto.edu /~yuana/AlgorithmManual/BOOK/BOOK4/NODE160.HTM (740 words)

	Cartan's Corner (using Charlotte Technology)
		The Cartan dynamical equations, describing the continuous topological evolution of physical systems whose topology may be encoded in terms of an exterior differential 1-form of Action, demonstrate that evolution in the direction of the topological torsion vector is thermodynamically irreversible on domains of topological (Pfaff) dimension of 2n+2.
		The topological thermodynamics of a very dilute van der Waals gas indicates that the universe could be modeled as a turbulent non-equilibrium open state of Pfaff topological dimension 4.
		Topological arguments are presented which yield results in agreement with Post's derivation, which was based on the Newmann principle of crystallographics, and in disagreement with the conventional dogma of Henley and Sakurai.
	www22.pair.com /csdc/car/carhomep.htm (2848 words)

	Encyclopedia article on Topological space [EncycloZine] (Site not responding. Last check: 2007-11-07)
		The sets in T are the open sets, and their complements in X are the closed sets.
		The attempt to classify the objects of this category by invariants has motivated and generated entire areas of research, such as homotopy theory, homology theory, and K-theory, to name just a few.
		A linear graph is a topological space that generalises many of the geometric aspects of graphs with vertices and edges.
	encyclozine.com /Topological_space (2350 words)

	Topological Science, Directory (Site not responding. Last check: 2007-11-07)
		Research Group on Topological Quantum Field Theory and Knots Research Group on Topological Quantum Field Theories in any dimension and their relation to topological invariants.
		Topological Geometrodynamics An attempt to unify fundamental interactions by assuming that physical spacetimes can be regarded as submanifolds of certain 8-dimensional space.
		Quantum Topology Project The objective of the Project are to use topological quantum field theories to explore low-dimensional topological objects.
	www.morrisarearedcross.org /bWFfMjEwMzE2.aspx (440 words)

	Topological Sort
		A topological sort of the vertices of G is a sequence
		Informally, a topological sort is a list of the vertices of a DAG in which all the successors of any given vertex appear in the sequence after that vertex.
		Clearly the first vertex in a topological sort must have in-degree zero and every DAG must contain at least one vertex with in-degree zero.
	www.brpreiss.com /books/opus6/html/page558.html (202 words)

	Shortest paths and topological ordering
		Here's another ordering that always works: define a topological ordering of a directed graph to be one in which, whenever we have an edge from x to y, the ordering visits x before y.
		If this algorithm terminates, L is a topological ordering, since we only add a vertex v when all its incoming edges have been deleted, at which point we know its predecessors are already all in the list.
		Incidentally this also proves that algorithm 6 finds a topological ordering whenever one exists, and that we can use algorithm 6 to test whether a graph is a DAG.
	www.ics.uci.edu /~eppstein/161/960208.html (1605 words)

	54: General topology (Site not responding. Last check: 2007-11-07)
		The most important functions between topological spaces are the continuous ones (a definition borrowed from analysis), which we use to define homeomorphisms -- functions which can be used to demonstrate that two spaces are "the same".
		One significant family of examples is sets S of functions between topological spaces X and Y. Depending on the properties or additional structures possessed by X and Y, S may be given one or more topologies, and in some cases itself possesses an additional structure.
		The study of topological vector spaces is treated in detail in functional analysis and related fields such as operator theory.
	www.math.niu.edu /~rusin/known-math/index/54-XX.html (2431 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.
		Two sets are said to be topologically equivalent if they are topological images of one another.
	www.cut-the-knot.org /do_you_know/topology.shtml (759 words)

	The Topology of Chaos: Alice in Stretch and Squeezeland
		Topological analysis is about extracting from chaotic data the topological signatures that determine the stretching and squeezing mechanisms which act on flows in phase space and are responsible for generating chaotic behavior.
		For three-dimensional systems, the methodology is well established and relies on sophisticated mathematical tools such as knot theory and templates (i.e., branched manifolds such as the one shown on the cover, see here for a color version with explanations).
		The last chapters discuss how topological analysis could be extended to handle higher-dimensional systems, and how it can be viewed as a key part of a general program for dynamical systems theory.
	www.thetopologyofchaos.net (1005 words)

	Topological Spaces (Site not responding. Last check: 2007-11-07)
		Atlas: On Mutual Compatificability of Topological Spaces by Martin Maria Kovar...
		Topological spaces - definition and basic concepts - Apronus.com...
		Topological Spaces: From Distance to Neighborhood by Buskes and van Rooij - Apro...
	www.scienceoxygen.com /math/528.html (152 words)

	Topological Zoo Home Page (Site not responding. Last check: 2007-11-07)
		The Topological Zoo is designed a resource for mathematicians and educators.
		The Topological Zoo is an ongoing project at the Geometry Center and is primarily the work of graduate students from the the University of Minnesota and undergraduates who participate in the Summer Institute at the Geometry Center.
		This exhibit describes the closed compact surfaces by their topological type, and includes the sphere, the torus, the Klein bottle, and the projective plane as some of its main attractions.
	www.geom.uiuc.edu /zoo (223 words)

	Topological Sort
		A topological sort of a DAG G = (V, E) is a linear ordering of v
		Topological sort can also be viewed as placing all the vertices along a horizontal line so that all directed edges go from left to right.
		Theorem: Topological-Sort (G) produces a topological sort of DAG, G. Proof: Run DFS (G) to determine finishing times.
	www.cs.fsu.edu /~cop4531/slideshow/chapter23/23-4.html (306 words)

	Topological Slide
		Mathematicians often describe the nature of certain topological surfaces by describing what it would be like to "take a walk" on the surface.
		The space will consist of a model of a topological surface to which the platform is bound and upon which it is free to slide.
		The Topological Slide premiered at the Art and Virtual Environments Symposium held in conjunction with the Fourth International Conference on Cyberspace at the Banff Centre in 1994.
	emsh.calarts.edu /~aka/topological_slide/Introduction.html (178 words)

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