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Topic: Topological group

Related Topics

Locally compact abelian group

General linear group

Lie group

Partition of unity

Locally compact space

Topological space

Haar measure

Uniform space

Characteristic class

Pseudometric space

Compact space

Hausdorff space

Harmonic analysis

Euclidean space

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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.
		The theory of group representations is almost identical for finite groups and for compact topological groups.
	en.wikipedia.org /wiki/Topological_group (804 words)

	Topological group
		In mathematics, a topological group G is a group which is also a topological space such that the group multiplication
		Important examples of non-abelian topological groups are given by the Lie groups (topological groups that are also manifolds), for instance by the group GL(n,R) of all invertible n-by-n matrices with real entries.
		An example of a topological group which is not a Lie group is given by the rational numbers Q.
	www.ebroadcast.com.au /lookup/encyclopedia/to/Topological_group.html (531 words)

	Topological abelian group - Wikipedia, the free encyclopedia
		In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group.
		That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative.
		The theory of topological groups applies also to TAGs, but more can be done with TAGs.
	en.wikipedia.org /wiki/Topological_abelian_group (113 words)

	Algebraic Topology: Homotopy
		Let X be a topological space that is the union of two path-connected subspaces A and B, where the intersection of A and B is nonempty and path-connected.
		Then the fundamental group of X is generated by (the images of) the fundamental groups of A and B.
		Theorem The group operation on X induces a group operation on P(X;x) that coincides with the old group operation, and P(X;x) is commutative.
	www.win.tue.nl /~aeb/at/algtop-3.html (2011 words)

	TAU Quantum Group - Topological Effects in Quantum Mechanics (Site not responding. Last check: 2007-11-07)
		Topological and geometrical effects are among the most striking quantum phenomena discovered since the creation of quantum mechanics in 1926.
		An analogous topological effect is the Aharonov-Casher (AC) effect [7], in which particles that carry a magnetic moment acquire a nontrivial phase while encircling a charged wire.
		Topological quantum phases are crucial in explaining superconductivity, the quantum Hall effect, the Josephson junction, flux quantization and many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behavior.
	www.tau.ac.il /~quantum/publicat/topo-effects.html (1118 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The proof is constructive, in that it explicitly constructs the group algebra as the enveloping C-algebra of the convolution algebra $L^1(G),$ and faithfully embeds the group* as unitaries in the multiplier algebra of the group algebra.
		it is not true for all topological groups that their continuous representation theory is isomorphic (in the sense of Gelfand--Raikov) to the representation theory of a C*--algebra.
		This generalisation of group algebras seem useful even for locally compact groups, because it allows the analysis of representation sets other than $\rsg.$ \item{(3)} For a small class of non-locally compact groups, $\sigma\hbox{--group algebras}$ were constructed for $\cl R.=\rsg$ in [Gr1].
	www.ma.utexas.edu /mp_arc/papers/04-118 (5822 words)

	Encyclopedia article on Topological space [EncycloZine] (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 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.
	encyclozine.com /Topological_space (2350 words)

	[No title]
		To a topological group G, we assign a naive G-spectrum DG, called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that DG detects Poincare du- ality in the sense that BG is a Poincare duality space if and only if DG is a homotopy finite spectrum.
		The procedure of killing homotopy groups shows that E can be ex- pressed up to homotopy as a filtered homotopy colimit of G-spectra Eff, where ff is an index and Effis a G-spectrum having a finite num- ber of (free) cells_in particular, the norm map is a weak equivalence for Eff.
		A group is a duality group (of dimension n) if and only if its dualizing spectrum D is unequivariantly weak equivalent to a Moore spectrum in degree -n on a torsion free abelian group.
	www.math.purdue.edu /research/atopology/Klein/quinn.txt (11324 words)

	22: Topological groups, Lie groups (Site not responding. Last check: 2007-11-07)
		Thus Lie groups and other topological groups lie at the convergence of the different areas of pure mathematics.
		Topological groups are covered well, Lie groups per se hardly at all.
		Connections among algebraicity, centers, and the fundamental group for Lie groups.
	www.math.niu.edu /~rusin/known-math/index/22-XX.html (348 words)

	PlanetMath: group cohomology (topological definition) (Site not responding. Last check: 2007-11-07)
		"group cohomology (topological definition)" is owned by whm22.
		Cross-references: isomorphic, divides, sphere, infinite, complex numbers, unit, connected, universal cover, covering map, linear maps, modules, universal, cells, quotient, fundamental group, sequence, homology, group, complex, structure, homotopy, classifying space, fibre map, quotient map, action, fixed point, contractible, topological group
		This is version 14 of group cohomology (topological definition), born on 2004-08-08, modified 2006-01-09.
	planetmath.org /encyclopedia/GroupCohomology3.html (161 words)

	M. Montserrat Bruguera
		A convergence group (G,\Xi), or briefly G, is a group for which the convergence structure \Xi is compatible with addition.
		If G is a LCA group, the continuous convergence structure in \Gamma G is precisely the convergence given by the compact open topology [3], thus, the "convergence dual" and the ordinary dual are identical.
		Topological abelian groups are, in an obvious way, convergence groups, therefore it is natural to compare reflexivity and BB-reflexivity for them.
	www.utm.edu /staff/jschomme/topology/c/a/a/h/10.htm (924 words)

	Topological Groups
		Topological groups is another rich source of interesting spaces.
		Topological groups form a particularly nice family of spaces, as it turns out that the group structure imposes severe restrictions on the topology.
		C we get the unitary groups U(n) and the special unitary groups SU(n), which is the subgroup of U(n) of matrices with determinant 1.
	www.maths.abdn.ac.uk /~ran/mx4509/mx4509-notes/node5.html (641 words)

	R&E 24 Abstracts
		We study the topological semigroups that admit the adjunction of a non-isolated absorbing element and the structure and permanence properties of the class AA of topological semigroups admitting this type of adjunctions.
		We observe that for a topological group G the following are equivalent: (i) every continuous action of G on a compact space is weakly almost periodic; (ii) G is precompact.
		The theory of Lie groups in finite dimensions relies on the strong connection between the two structures of the group: the finite dimenional manifold structure and the topological group structure.
	www.heldermann.de /R&E/rae24abs.htm (1668 words)

	Further Results (Site not responding. Last check: 2007-11-07)
		The class of all topological groups which are topologically isomorphic to a subgroup of a Banach space is the class of all abelian topological groups G(with identity e) that admit a metric d which has the property that for all
		The class of all topological groups which are topologically isomorphic to a quotient group of a subgroup of a product of Banach spaces is the class of all abelian topological groups (Corollary 3.4).
		The class of all topological groups (respectively, Hausdorff topological groups) topologically isomorphic to a quotient group of a subgroup of a Banach space is the class of all pseudo-metrizable abelian topological groups (respectively, metrizable abelian topological groups).
	cedir.uow.edu.au /Projects/math_test/node4.html (625 words)

	The Variety Generated by Banach Spaces (Site not responding. Last check: 2007-11-07)
		Every Hausdorff topological group is a quotient space of a topological space which 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.
		Corollary 3.4 The variety of topological groups generated by the class of all topological groups that underlie Banach spaces is exactly the variety of all abelian topological groups.
	cedir.uow.edu.au /Projects/math_test/node3.html (755 words)

	Group Theory Glossary (Site not responding. Last check: 2007-11-07)
		A group is a set together with a method of combining elements to get new ones (addition or multiplication or...) which satisfies certain properties making it suitable for a wide variety of applications.
		Any locally compact Hausdorff topological group has a unique (up to scalars) nonzero left invariant measure which is finite on compact sets.
		A topological group is a set which has both the structure of a group and of a topological space in such a way that the operations defining the group structure give continuous maps in the topological structure.
	www.emba.uvm.edu /~jge/library/glossary/group_glossary.htm (144 words)

	HogBlog: Fundamental Groups of Topological Groups are Abelian
		Topologically these are great because they're compact smooth (even real-analytic) manifolds, and things like their fundamental groups are easy to extract from (for SO and SU) the fibrations we get by considering their actions on spheres.
		One compact topological group that isn't a Lie group is the p-adic integers, for any prime p: they actually form a compact topological ring.
		Another example of compact topological groups would be compact linear algebraic groups: consider the orthogonal group of nxn matrices over an arbitrary field k, even of prime characteristic, and give it the Zariski topology, making it an affine variety.
	www.koschei.net /blog/archives/000434.html (984 words)

	Topological Vector Space
		A topological vector space is a vector space with a topology, such that addition and scaling are continuous.
		A normed vector space is a topological vector space, deriving its topology from the metric.
		Let s be a topological group with a local base at 0.
	www.mathreference.com /top-ban,tvs.html (1410 words)

	OctoberNovember2004
		ABSTRACT: A group with topology (X,,T) is called a `semitopological group' if multiplication is separately continuous, a `paratopological group' if multiplication is jointly continuous and, of course, a `topological* group' if inversion is continuous as well.
		In two classic papers of 1957, Robert Ellis proved, for locally compact Hausdorff spaces, that every paratopological group is a topological group and then, a few months later, that every semitopological group is a topological group.
		The focus is on developing computer-aided methods for determining the stability and consistency of computational representations of geometric objects in the context of the topological, piecewise linear, embedding, and differential topological categories, among others.
	topann.home.att.net /OctNov2004.html (823 words)

	Amazon.com: Topological Methods in Hydrodynamics (Applied Mathematical Sciences): Books: Vladimir I. Arnold,Boris A. ... (Site not responding. Last check: 2007-11-07)
		Topological hydrodynamics is a young branch of mathematics studying topological features of flows with complicated trajectories, as well as their applications to fluid motions.
		Topological Methods in Hydrodynamics is the first monograph to treat topological, group-theoretic, and geometric problems of ideal hydrodynamics and magnetohydrodynamics for a unified point of view.
		The group we will most often be dealing with in hydrodynamics is the infinite-dimensional group of diffeomorphisms that preserve the volume element of the domain of a fluid flow.
	www.amazon.com /exec/obidos/tg/detail/-/038794947X?v=glance (639 words)

	PlanetMath: topological group (Site not responding. Last check: 2007-11-07)
		is continuous on G. "topological group" is owned by Evandar.
		This is version 3 of topological group, born on 2002-01-22, modified 2002-10-19.
		(Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Structure of general topological groups)
	planetmath.org /encyclopedia/TopologicalGroup.html (64 words)

	[No title]
		Then $G$, together with this multiplication on the left by elements of $N$, is referred to as a topological $N$-group or topological nearmodule.
		In this paper, $N$ will be a topological nearring whose additive group is a Euclidean $N$-group and $G$ will be the additive group $R$ of real numbers.
		It is proved that a positive definite function on the semigroup of the positive elements of a separable function space is the Laplace transform of a measure concentrated on the set of positive functionals.
	web.math.hr /~glasnik/vol_31/gl96-1.txt (729 words)

	Group actions
		In algebra, geometry and topology we often exploit the fact that important structures arise from families of morphisms that are indexed by a group.
		For example, rotations in the plane about the origin are indexed by the unimodular group of complex numbers; we say that this group acts on the plane and the orbit of a point at distance r from the origin is the circle of radius r.
		G is a Lie group, so G has a differentiable structure with respect to which its binary operation and the taking of inverses is smooth, and X is a smooth manifold.
	www.ma.umist.ac.uk /kd/curves/node4.html (797 words)

	TOPOLOGICAL METHODS IN GROUP THEORY
		First, it is for graduate students who have had an introductory course in algebraic topology and who need a bridge from common knowledge to the current research literature in geometric and homological group theory.
		Secondly, I am writing for group theorists who would like to know more about the topological side of their subject but who have been too long away from topology.
		5.12 Compactifiability at infinity as a group invariant
	www.math.binghamton.edu /ross/contents.html (289 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Vladimir Pestov, "Free abelian topological groups and the free locally convex space on the unit interval", J. London Math.
		[85] with Sheila Oates-Williams, "A characterization of the topological group of p-adic integers", Bull.
		[60] with Eli Katz "On endomorphisms of abelian topological groups", Proc.
	uob-community.ballarat.edu.au /~smorris/publ.htm (2053 words)

	Michael G. Tka\v{c}enko (Site not responding. Last check: 2007-11-07)
		It is well known that every compact connected Abelian group of weight less than or equal to the continuum is monothetic, i.
		For a topological group G, let b(G) be the least cardinal \kappa such that for every neighborhood of the identity in G there exists a subset C \subseteq G with C Theorem 2.
		A separable totally bounded topological group of countable pseudocharacter has a suitable set.
	www.utm.edu /staff/jschomme/topology/c/a/a/l/19.htm (329 words)

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