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Topic: Basis (topology)

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Topology glossary

product topology

closure topology

completeness topology

Zariski topology

base topology

continuity topology

differential topology

weak topology

general topology

neighborhood topology

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	[No title]
		Topology Glossary Mainly extracted from (a) UC Davis Math:Profile Glossary (http://www.math.ucdavis.edu/profiles/glossary.html) by Greg Kuperberg (http://www.math.ucdavis.edu/profiles/kuperberg.html), and (b) Topology Atlas Glossary (http://www.achilles.net/~mtalbot/TopoGloss.html).
		In mathematics, usually means basis in the sense of linear algebra; a minimal set of vectors that spans a vector space.
		An early result in topology states that every closed 3-manifold (closed meaning that the manifold is finite and connected but has no boundary) has a Heegaard splitting and a resulting description in terms of a Heegaard diagram, which describes how the two handlebodies are glued together.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	Product topology - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.
		In general, the box topology is finer than the product topology, but for finite products they coincide.
		The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology.
	en.wikipedia.org /wiki/Product_topology (803 words)

	[No title]
		The topology P consists of the elements in O and the complements of closed subsets as neighborhoods of that point.
		Hausdorff topology +------------------------------------------------------------ The Hausdorff topology is a metric on the set of closed bounded subsets of a complete metric space.
		induced topology +------------------------------------------------------------ The induced topology on a subset A of X, where (X,T) is a topological spoace is the the topological space (A, Y cap A _Y in T).
	www.math.harvard.edu /~knill/sofia/data/topology.txt (1652 words)

	[No title]
		Section 4 defines the notion of sheaf on either a topology or a basis and culminates wi* th the proof in our cases that sheaves on a basis* agree with sheaves on the topolo* gy generated by the basis*.
		C(?; C) a cover in the topology precisely when there is an element of the basis R0 such t* hat R0is a subobject of R. Proof.Since pullbacks preserve subobjects by definition, the first two axioms f or a Grothendieck topology* are obvious from those for a basis.
		Suppose the subcategory of sheaves on a basis is reflective.
	hopf.math.purdue.edu /JohnsonM/shfloop.txt (7069 words)

	Basis - Wikipedia, the free encyclopedia
		Look up basis on Wiktionary, the free dictionary.
		In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects.
		In economics or accounting, a basis is the cost or value of an asset as adjusted for tax purposes.
	en.wikipedia.org /wiki/Basis (119 words)

	Algebraic Topology: Topology
		A basis for the open sets is a collection of open sets such that each open set is a union of some subcollection.
		The topology on A defined by F is the weakest topology (i.e., the smallest collection OA) for which all these functions become continuous.
		The topology on B defined by F is the strongest topology (i.e., the largest collection OB) for which all these functions become continuous.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	World Intellectual Property Organization (Site not responding. Last check: 2007-11-06)
		The topology of the decoding logic wherein each logic unit of the first kind comprises three NAND-gates is functionally equivalent with the topology of the known decoding logic wherein each logic unit of the first kind comprises a known EXOR-gate.
		However, the topology of decoding logic according to the invention is matched with advantageous logic hardware components and thus can operate considerably faster than the known decoding logic.
		This is a consequence of the fact that an EXOR-gate has a relatively complex topology as a consequence of which it is a relatively slow operating logical hardware component.
	www.wipo.int /ipdl/IPDL-CIMAGES/view/pct/getbykey5?KEY=03/107538.031224&ELEMENT_SET=DECL (2979 words)

	FreeTree - freeware program for construction of phylogenetic trees (Site not responding. Last check: 2007-11-06)
		In the present time the methods of construction of phylogenetic trees on the basis of molecular data are widely used not only systematic and comparative biology, but also in ecology, ethology, sociobiology and epidemiology.
		While all published trees constructed on the basis of DNA sequencing presently contain the information about the robustness of the tree topology (mostly the bootstrapping values for internal branches of the tree) this fundamental information is never provided for the trees constructed on the basis of multi-loci methods.
		When we intend to construct the tree on the basis of nucleotide distances computed from fractions of shared restriction fragments by the Nei and Li iterations method {5326} we must add a second empty row with the length of a recognition site of a particular restriction enzyme written in its first cell.
	www.natur.cuni.cz /asc/~flegr/freetree.htm (1418 words)

	NTU Info Centre: Topology glossary (Site not responding. Last check: 2007-11-06)
		The topology T is the smallest topology on X containing B and is said to be generated by B.
		Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
		The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
	www.nowtryus.com /article:Partition_of_unity (4072 words)

	Exercises 6
		The closure of U in the subspace topology on A is equal to the closure of U in the topology on X.
		Prove that the set of all unbounded open intervals of R forms a sub-basis for the usual topology on R which is not a basis.
		Prove that the discrete topology on R does not have a countable basis.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Tutorials/T6.html (357 words)

	Topology glossary : Basis (topology) (Site not responding. Last check: 2007-11-06)
		This is a glossary of some terms used in the branch of mathematics known as topology.
		From the standpoint of topology, X and Y are the same.
		A set of open sets is a sub-base for a topology if every open set is a union of finite intersections of sets in the sub-base.
	www.city-search.org /ba/basis-(topology).html (1299 words)

	Topology correlation and test preparation
		Topology correlation is the phase where one correlates test and model geometrical and sensor/shaker configurations.
		Note that when defining test nodes in a local basis, the node selection commands are applied in the global coordinate system.
		The most common source of problems with the topology correlation commands is the use of models with nodes not attached to any element.
	www.sdtools.com /help/topo.html (865 words)

	wang-cevpn-group-00.txt
		For other topology such as a hub-spoke VPN, a packet may have to traverse more than one IPsec tunnel to reach its destination.
		4.2 Complex Topology Decomposition A small and simple network can be designed and managed using just one VPN group, in the form of a full mesh, a partial mesh or a hub- and-spoke topology.
		For a hub-spoke topology, usually the spoke router selects the hub router as the default gateway and hub router provides inter- spoke connection.
	ietfreport.isoc.org /idref/draft-wang-cevpn-group (2975 words)

	James Tauber : Poincare Project: A Basis for a Topology
		You can characterise a topology by describing a certain class of open sets from which the other open sets can be calculated.
		Because members of the basis are themselves open sets, once we have a basis we can generate all the other open sets by taking unions.
		This, along with the requirement that every element must appear in at least one basis open set is sufficient to ensure that one has a basis for a topology.
	www.jtauber.com /blog/2004/12/10/poincare_project:_a_basis_for_a_topology (311 words)

	Topology (Site not responding. Last check: 2007-11-06)
		Topology forms the basis for many subdisciplines of Mathematics, including Real and Complex Analysis and Fuctional Analysis.
		The pre-requisite for this class is the knowledge of the topology of the real line as taaught in a standard upper division analysis class.
		Students, who are not familiar with the notions of intuitive set theory are asked to study this on their own in the first few weeks of classes.
	www.csun.edu /~hcmth017/math501.html (293 words)

	Topology II, Spring 2005 -- Course Information
		This is the second semester of a one year course designed to provide a modern introduction to Topology.
		In this second semester, we will spend most of our time on algebraic topology, but there will be applications from both general and differntial topology.
		Semester grades will be determined on the basis of classwork (including both homework and class participation), (at least one) major examination, and a final exam, all equally weighted.
	www.math.tamu.edu /~jon.pitts/courses/2005a/637/information.html (307 words)

	Radial Basis Function (Site not responding. Last check: 2007-11-06)
		Radial basis function (RBF) networks have a static Gaussian function as the nonlinearity for the hidden layer processing elements.
		Any supervised topology (such as a MLP) may be used for the classification of the weighted input.
		The advantage of the radial basis function network is that it finds the input to output map using local approximators.
	www.nd.com /models/rbf.htm (255 words)

	A non-Hausdorff interval with two right ends (from topology) -- Encyclopædia Britannica (Site not responding. Last check: 2007-11-06)
		Let pq be a closed interval with the ordinary linear topology.
		Basis elements are either open intervals or half-open intervals that contain either p or q.
		A basis element is either a basis element of the original interval pq or the union…
	www.britannica.com /eb/article-69117?tocId=69117 (99 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		It is assumed that students have a working knowledge of the equivalent of a one semester course in general topology (for example, see the appended syllabus for the undergraduate course M367K).
		For the semester in differential topology, it will also be assumed that students know the basic material from an undergraduate linear algebra course.
		Milnor, Topology from the Differentiable Viewpoint, University of Virginia Press, 1965.
	www.ma.utexas.edu /Prelims-Syllabi/topology.html (280 words)

	Topology in Computer Science
		Let B be a basis of open sets for X, closed under finite intersections.
		The difficulty lying in part 1 is illustrated by the following example: Let X be the four-point lattice {a, b, c, d} with least element a, greatest element d, and two incomparable elements b and c in between.
		A basis for the Scott topology on X is given by B = {A, B, C, D where A = {a,b,c,d}, B = {b,d}, C = {c,d}, and D = {d}.
	www.informatik.uni-siegen.de /theo/TopCS.html (1549 words)

	Topology
		The space X is said to be second countable if there exists a countable basis of the open sets.
		the induced topology is exactly the topology one obtains by viewing S as a metric space, where the metric is inherited from the metric space
		As a basis for the topology one may thus choose the sets
	www.math.ku.dk /~jakobsen/uge/to/to_en.html (185 words)

	► » Topology (Site not responding. Last check: 2007-11-06)
		It is well known that the open balls are a basis for a metric
		a topology it is necessary that the intersection of any two can
		Yes, because, by definition, every open set is an union of open disks.
	www.science-chat.org /Topology-6931373.html (619 words)

	Topology I, Fall 2003 -- Course Information (Site not responding. Last check: 2007-11-06)
		This is the first semester of a one year course designed to provide a modern introduction to Topology.
		In General Topology, we cover the basic notions of topological spaces and subspaces and their basic attributes, such as:
		In this first semester, we should cover all of the General Topology and some of the Differential Topology.
	www.math.tamu.edu /~jon.pitts/courses/2003c/636/information.html (280 words)

	basis - OneLook Dictionary Search
		Basis : 101 Investment Terms You Should Know [home, info]
		Phrases that include basis: basis point, cost basis, adjusted basis, basis risk, cash basis, more...
		Words similar to basis: foundation, groundwork, base, bases, cornerstone, footing, fundament, ground, grounds, reason, root, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=basis (426 words)

	► » what sets admit connected Hausdorff topological spaces? (Site not responding. Last check: 2007-11-06)
		I take it that this topology is defined on the set X = N \ {0}.
		k is in the basis set { k + p*n, n in N}.
		Steen and Seebach "Counterexamples in Topology", examples 60 and 61.
	www.science-chat.org /what-sets-admit-connected-Hausdorff-topological-spaces-7227812.html (1215 words)

	Amazon.com: Books: An Introduction to Algebraic Topology (Graduate Texts in Mathematics) (Site not responding. Last check: 2007-11-06)
		A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) by J.
		It is suitable for a two- semester course at the beginning graduate level, requiring as a prerequisite a knowledge of point set topology and basic algebra.
		One expects algebraic topology to be a mixture of algebra and topology, and that is exactly what it is. Read the first page
	www.amazon.com /exec/obidos/tg/detail/-/0387966781?v=glance (688 words)

	Review for Mid-term (Site not responding. Last check: 2007-11-06)
		Know what a topological space is. Given a set and a collection of subsets of the set, determine whether the given collection of subsets forms a topology on the set.
		Given a collection of sets, be able to verify that the sets form the basis for some topology, and be able to determine which sets are open in the topology with that basis.
		Be able to decide whether two bases generate the same topology on a given set X. Know which subsets of R
	www.tcnj.edu /~ccurtis/classes/f02405midtermreview.htm (374 words)

	RBF++ : A library for Radial Basis Functions
		In contrast to classic multi layer perceptrons (see [2] for example) the activation of a neuron is not given by the weighted sum of all its inputs but by the computation of a radial basis function.
		--> is used where x is the input of the neuron, mu is the basis of the neuron, and sigma is the amplitude of the neuron.
		RBF-Networks are feed forward run and consist of one input layer (x), one hidden layer of Gaussian neurons (G), and one output layer (o).
	wwwradig.in.tum.de /people/buck/RBF (630 words)

	[No title]
		a basis for a topology T of a set X is a collection of subsets of X (called basis elements) such that
		the indiscrete topology of a set X is the topology which consists of the empty set and the set \ $X$.
		Then the quotient topology of Y is the collection of all subsets of Y whose preimage under f is open in X. back)
	www.math.uni-frankfurt.de /~johannso/Vorlesung/HTML/definition.html (1019 words)

	Geometry & Topology Publications (Site not responding. Last check: 2007-11-06)
		GTP is based in the Mathematics Department of the University of Warwick at Coventry, UK.
		The purpose of all Geometry and Topology Publications is the advancement of mathematics.
		Editors evaluate submissions strictly on the basis of scientific merit, without regard to authors' nationality, country of residence, institutional affiliation, sex, ethnic origin and political views.
	www.maths.warwick.ac.uk /gt/gtp.html (146 words)

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