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	math lessons - Topological ring
		In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps
		Topological rings occur in mathematical analysis, for examples as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space; all Banach algebras are topological rings.
		Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.
	www.mathdaily.com /lessons/Topological_ring (364 words)

	Encyclopedia :: encyclopedia : Ring theory (Site not responding. Last check: 2007-09-17)
		In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		The study of rings originated from the theory of polynomial rings and the theory of algebraic integers.
		A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces.
	www.hallencyclopedia.com /topic/Ring_theory.html (734 words)

	Topological ring
		In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R
		Topological rings occur in mathematical analysis, for examples as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space; all Banach algebras are topological rings.
		Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.
	www.ebroadcast.com.au /lookup/encyclopedia/i-/I-adic_topology.html (301 words)

	NationMaster - Encyclopedia: Topological vector space
		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 topological vector space X is a vector space over a topological field K (most often the real or complex numbers with their standard topologies) which is endowed with a topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous functions.
	www.nationmaster.com /encyclopedia/Topological-vector-space (3637 words)

	PlanetMath: topological ring
		which is a topological space is called a topological ring if the addition and multiplication functions are continuous functions from
		A topological ring is defined to be a topological field if it is a field and the inversion function is continuous away from 0.
		This is version 2 of topological ring, born on 2002-06-08, modified 2006-06-30.
	planetmath.org /encyclopedia/TopologicalRing.html (141 words)

	TOPOLOGICAL GROUPS : Encyclopedia Entry
		In mathematics, a topological group is a group G together with a topology on G such that the group and topological structures are compatible.
		In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets.
		An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces.
	www.bibleocean.com /OmniDefinition/Topological_groups (1031 words)

	Topological group (Site not responding. Last check: 2007-09-17)
		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.
		Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.
		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.
	www.punweb.com /article/Topological_group (777 words)

	Topological_space - The Wordbook Encyclopedia (Site not responding. Last check: 2007-09-17)
		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.
		Topological spaces can be broadly classified, up to homeomorphism, by their topological properties.
		Topological spaces provide the most common notions of closeness and convergence for a space, but it may be possible in some cases to study more specialized or more general notions.
	www.thewordbook.com /Topological_space (1885 words)

	Topological Rings
		Imagine the object above in the figure to the left made from perfectly elastic materical.
		One possible way is to cut one ring, move the other ring through the gap, and rejoin the the first ring exactly as it was.
		However, it is also possible to transform the first shape into the second without any cutting, simply by manipulating the objects in the appropriate manner (stretching, bending, but not breaking).
	people.reed.edu /~mcphailb/puzzles/loops.html (97 words)

	Topological ring
		Topological rings occur in mathematical analysis, forexamples as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or asrings of continuous linear operators on some normed vector space ; all Banach algebras are topological rings.
		Every topological ring is a topological group (with respect toaddition) and hence a uniform space in a natural manner.
		One can thus askwhether a given topological ring R is complete.If it is not, then it can be completed: one can find an essentially unique complete topological ring S whichcontains R as a dense subring such that the given topology on R equals the subspace topology arising from S.
	www.therfcc.org /topological-ring-220320.html (303 words)

	PlanetMath: topological ring
		which is a topological space is called a topological ring if the addition and multiplication functions are continuous functions from
		A topological ring is defined to be a topological field if it is a field and the inversion function is continuous away from 0.
		This is version 2 of topological ring, born on 2002-06-08, modified 2006-06-30.
	www.planetmath.org /encyclopedia/TopologicalRing.html (141 words)

	PlanetMath: normed algebra
		In addition, the last condition ensures that the ring multiplication is continuous.
		A normed algebra that is complete with respect to the norm is called Banach algebra (the underlying field must be complete and algebraically closed), paralleling with the analogy with a Banach space versus a normed vector space.
		Normed rings and normed algebras are special cases of the more general notions of a topological ring and a topological algebra, the latter of which is defined as a topological ring over a field such that the scalar multiplication is continuous.
	www.planetmath.org /encyclopedia/RealNormedAlgebra.html (278 words)

	Definition of index.php?search=class\|ring&limit=20&offset=60 (Site not responding. Last check: 2007-09-17)
		ring]] from a given ring and a [[monoid]].
		2:...the left ring finger because the vein in the left ring finger is directly connected to the [[heart]], a...
		1:...], a '''topological ring''' is a [[ring (algebra)
	www.wordiq.com /knowledge/index.php?search=class%7Cring&limit=20&offset=60 (846 words)

	Cartan's Corner (using Charlotte Technology)
		The topological defects, in the otherwise flat surface of fluid density discontinuity, appear as a pair of zero mean curvature surfaces, with a conical (dimple) singularity at each end.
		It follows that topological fluctuations in kinematics are generated by processes that have spinor components.
		The Falaco Solitons are topologically universal phenomena created experimentally by a rotational dynamics in a continuous media with a discontinuity surface, such as that found in a swimming pool.
	www22.pair.com /csdc/car/carhomep.htm (1630 words)

	Topological ring (Site not responding. Last check: 2007-09-17)
		Topological rings occur in mathematical analysis for examples as rings of continuous functions on some topological space (where the is given by pointwise convergence) or as of continuous linear operators on some normed vector space ; all Banach algebras are topological rings.
		The rational real complex and p -adic numbers are also topological rings (even fields) with their standard topologies.
		The rings of formal power series and the p -adic integers are most naturally defined as completions certain topological rings carrying I -adic topologies.
	www.freeglossary.com /Topological_ring (439 words)

	Definition of index.php?search=ring&limit=20&offset=80 (Site not responding. Last check: 2007-09-17)
		Ring species illustrate that the species concept is no...
		From the standpoint of ring theory, isomorphic rings cannot be distinguished.
		rings originated from the theory of [[polynomial ring]]s and the theory of [[algebraic integer]]s.
	www.wordiq.com /knowledge/index.php?search=ring&limit=20&offset=80 (886 words)

	GardeningDaily - Topological group
		Here, G × G is viewed as a topological space by using the product topology.
		In fancier language, one can say that topological groups are group objects in the category of topological spaces.
		If H is a subgroup of G the set of left or right cosets G/H is a topological space when given the quotient topology (the finest topology on G/H which makes the natural projection q : G → G/H continuous).
	www.gardeningdaily.com /flowers-and-plants/Topological_group (786 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-09-17)
		Thus, since the union of the empty collection is the empty set, and the intersection of the empty collection is (by convention) X, an equivalent definition can be given by requiring that a topology be closed under unions and finite intersections.
		If Γ is an ordinal number, then the set Γ may be endowed with the order topology generated by the intervals (a, b), where a and b are elements of Γ.
		Topological Space at ProvenMath strict definitions and formal proofs.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=topological_space (1935 words)

	topological ring - OneLook Dictionary Search (Site not responding. Last check: 2007-09-17)
		We found 3 dictionaries with English definitions that include the word topological ring:
		Tip: Click on the first link on a line below to go directly to a page where "topological ring" is defined.
		Topological Ring : Eric Weisstein's World of Mathematics [home, info]
	www.onelook.com /?loc=rescb&w=topological+ring (81 words)

	Topological dimension
		It was a triumph of early 20th century mathematics to finally give a precise definition of topological dimension.
		In a topological space X, we do not assume that we have a distance function, but we do assume that we know what the open subsets are.
		Some references for both topological and fractal dimensions are [Edg90, Fal90].
	www.math.okstate.edu /mathdept/dynamics/lecnotes/node36.html (749 words)

	Highbeam Encyclopedia - Search Results for Topological ring
		Titanium wedding rings Tungsten and titanium wedding rings with gold or platinum inlays.
		Variable-scale topological data structures suitable for progressive data transfer: the GAP-face tree and GAP-edge forest.(Generalized Area Partitioning)
		Affinity chromatography--from textile dyes to synthetic ligands by design, Part I: very significant progress has been made since the mid-1970s when dye ligands were first introduced.
	www.encyclopedia.com /SearchResults.aspx?Q=Topological+ring (359 words)

	Topological module - Encyclopedia, History, Geography and Biography
		Topological module - Encyclopedia, History, Geography and Biography
		In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
		This encyclopedia, history, geography and biography article about Topological module contains research on
	www.arikah.com /encyclopedia/Topological_module (66 words)

	Bounded set (topological vector space) - Article from FactBug.org - the fast Wikipedia mirror site
		Bounded sets are a natural way to define a locally convex polar topologies on the vector spaces in a dual pair, as the polar of a bounded set is an absolutely convex and absorbing set.
		Given a topological vector space (X,τ) over a field F, S is called bounded if for every neighborhood of the zero vector N there exists a scalar α so that
		A subset A of a topological module M over a topological ring R is bounded if for any neighborhood N of o
	www.factbug.org /cgi-bin/a.cgi?a=1789812 (353 words)

	Category "Topological groups" - Encyclopedia, History, Geography and Biography
		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.
		There are 4 subcategories shown below (more may be shown on subsequent pages).
		Category "Topological groups", Topology, Group theory and Harmonic analysis.
	www.arikah.com /encyclopedia/Category/Topological_groups (110 words)

	Mathematical Subject Classification
		Associative rings and algebras {For the commutative case, see 13-xx}
		Category theory; homological algebra {For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for algebraic topology}
		Topological groups, Lie groups {For transformation groups, see 54H15, 57Sxx, 58-xx.
	www.ams.org /msc (607 words)

	topological module - Information from Reference.com
		Since every subset of a topological module over a discrete ring is...
		In mathematics, a topological module is a module over a topological ring such
		Thus, linearly-compact modules in the narrow sense are the topological...
	www.reference.com /search?q=topological%20module&db=web (278 words)

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