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Topic: Topology

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List of general topology topics

Geometric topology

Connectedness

Banach norm

Compact space

Topologies on the set of operators on a Hilbert space

Algebraic topology

Topology of the universe

General topology

Weak topology

Topological neighbourhood

Topological space

Metric space

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	Topology - Wikipedia, the free encyclopedia
		Topology has sometimes been called rubber-sheet geometry, because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing).
		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.
		However, Lacan's use of topology, like his use of algebra, does not meet the standards of rigour normally evinced by a mathematical discipline, and should be seen more as an analogy (the value of which is left to the reader to decide upon) than as a branch of applied mathematics.
	en.wikipedia.org /wiki/Topology (1876 words)

	ESRI News -- ArcNews Summer 2002 Issue -- ArcGIS 8.3 Brings Topology to the Geodatabase
		Topology rules, when applied to geographic features or feature classes in a geodatabase, enable GIS users to model such spatial relationships as connectivity (are all of my road lines connected?) and adjacency (are there gaps between my parcel polygons?).
		Topology is also used to manage the integrity of coincident geometry between different feature classes (e.g., are the coastlines and country boundaries coincident?).
		A topology is a set of integrity rules for the spatial relationships along with a few important properties: a cluster tolerance, feature class ranks (for coordinate accuracy), errors (rule violations), and any exceptions to the rules you've defined.
	www.esri.com /news/arcnews/summer02articles/arcgis83-brings.html (2242 words)

	[No title]
		Topology is a math discipline who treats of the research of invariants in a geometry cleared of all idea of measurement or of distance.
		Topology is a branch of pure mathematics, deals with the fundamental properties of abstract spaces.
		Topology is also concerned with the ways in which one manifold may be embedded within another, such as the ways a knotted circle may be embedded in three-dimensional space.
	www.chez.com /alcochet/toposi.htm (1679 words)

	Topology - Uncyclopedia
		Topology is a particularly virulent, yet fortunately rare, strain of mathematics.
		The most common symptom of topology is confusing everyday objects with one another, such as being unable to differentiate between doughnuts and coffee cups, which obviously causes many problems for police officers.
		Topology was discovered by Leonhard Euler, the town drunkard of Königsberg, when he attempted to find his way home after a particularly long night of heavy drinking.
	uncyclopedia.org /wiki/Topology (255 words)

	Topology
		Topology is the mathematical study of those properties that are preserved through continuous deformations of objects.
		A circle is topologically equivalent to an ellipse, a sphere is equivalent to a cube, and a coffee cup to a donut.
		Geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups.
	www.math.neu.edu /research/topology.html (257 words)

	Category:Topology - Wikipedia, the free encyclopedia
		In mathematics, topology is a branch of geometry concerned with the study of topological spaces.
		See the topology glossary for common terms and their definition.
		Properties of general topological spaces (as opposed to manifolds) are discussed in general topology.
	en.wikipedia.org /wiki/Category:Topology (136 words)

	54: General topology
		Topology is the study of sets on which one has a notion of "closeness" -- enough to decide which functions defined on it are continuous.
		Thus a general theme in topology is to test the extent to which the axioms force the kind of structure one expects to use and then, as appropriate, introduce other axioms so as to better match the intended application.
		Since the axioms of topology are stated in terms of subsets of X, it should be no surprise that one branch of topology is closely related to set theory, particularly "descriptive set theory".
	www.math.niu.edu /~rusin/known-math/index/54-XX.html (2431 words)

	[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).
		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.
		In 3-dimensional topology, a surface in a 3-manifold with the property that no essential circle in the surface bounds a disk in the manifold.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	topology. The Columbia Encyclopedia, Sixth Edition. 2001-05
		Topology is sometimes referred to popularly as “rubber-sheet geometry”; because a figure can be changed to that of an equivalent figure by bending, stretching, twisting, and the like, but not by tearing or cutting.
		Topology is concerned with those properties of geometric figures that are invariant under continuous transformations.
		A surface is a simple example of a topological space, the basic entity studied in topology.
	www.bartleby.com /65/to/topology.html (892 words)

	PlanetMath: topological space
		Can a topology be defined as a subset of an arbitrary complete (and complemented) lattice, instead of a power set?
		it might be interesting to note that you can equivalently define a topology in terms of it's closed sets, by demorgan's set laws.
		i can't seem to find a definition for the topology induced by a metric space either, which is something you probably want to add.
	planetmath.org /encyclopedia/Topology.html (300 words)

	Math Forum: Leonard Euler and the Bridges of Konigsberg
		Topology is one of the newest branches of mathematics.
		A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.
		The foundations of topology are often not part of high school math curricula, and thus for many it sounds strange and intimidating.
	mathforum.org /isaac/problems/bridges1.html (336 words)

	Topology - Wikibooks, collection of open-content textbooks
		General Topology is based solely on set theory and concerns itself with structures of sets.
		Topology is a generalization of analysis and geometry.
		A naive description of topology is that it identifies those qualities of a space that do not change under twisting and stretching of that space.
	en.wikibooks.org /wiki/Topology (395 words)

	Topology - MSN Encarta
		Topology, branch of mathematics that explores certain properties of geometrical figures.
		In 1930, the word topology was coined by mathematician...
		Technology: Cyberspace is a topology, not a topography.…
	encarta.msn.com /encyclopedia_761558984/Topology.html (46 words)

	Chapter 5: Topology
		Physical topology should not be confused with logical topology which is the method used to pass information between workstations.
		A star topology is designed with each node (file server, workstations, and peripherals) connected directly to a central network hub or concentrator (See fig.
		A star-wired ring topology may appear (externally) to be the same as a star topology.
	fcit.coedu.usf.edu /network/chap5/chap5.htm (718 words)

	topology - a Whatis.com definition
		In the star network topology, there is a central computer or server to which all the workstations are directly connected.
		In the ring network topology, the workstations are connected in a closed loop configuration.
		In the partial mesh topology, some workstations are connected to all the others, and some are connected only to those other nodes with which they exchange the most data.
	searchnetworking.techtarget.com /gDefinition/0,,sid7_gci213156,00.html (420 words)

	Topology
		The family t is called a topology (for X) when it satisfies these axioms and its elements are called _open sets_ (open wrt the topology).
		The reader should now check that continuity in the sense of calculus of a function from R to R is equivalent to continuity as a map of topological spaces, with respect to the topology m.
		Call a topology t _stronger_ than the topology t' (both for the same set X) if t is contained in t'.
	www.georgetown.edu /faculty/kainen/topology.html (1132 words)

	An Atlas of Cyberspaces - Topology Maps
		The evolving topology of SuperJANET, the high-speed academic network in the UK.
		The map above left shows the ATM network circa September 1996, while the map on the right is a schematic map of the backbone topology of SuperJANET III from early 1998.
		The topology map below shows the new SuperJANET4 backbone, as of March 2001.
	www.cybergeography.org /atlas/topology.html (668 words)

	Understanding Topology and Shapefiles
		One of the primary reasons topology was developed was to provide a rigorous, automated method to clean up data entry errors and verify data.
		The typical digitizing procedure is to digitize all lines, build topology, and label polygons and then clean up slivers, dangles, and under- and overshoots and build topology again, repeating the clean and build phases as many times as necessary.
		The first step in enforcing planar topology in a shapefile is to remove twisted or self-intersecting polygon rings and to ensure that the "inside" of the polygon is on the correct side of the polygon boundary.
	www.esri.com /news/arcuser/0401/topo.html (1912 words)

	55: Algebraic topology
		Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants reflect some of the topological structure of the spaces.
		Apart from homology groups and their kin, the principal algebraic tool used in topology is the set of homotopy groups of a space, and related concepts; in particular this includes the fundamental group (pi_1(X)) of a space.
		The tools of algebraic topology, when developed in isolation or for applications to other fields such as ring theory, give rise to homological algebra and category theory; this is the proper framework for comparing different algebraic tools.
	www.math.niu.edu /~rusin/known-math/index/55-XX.html (2581 words)

	Open Directory - Science: Math: Topology (Site not responding. Last check: 2007-10-22)
		Algebraic Topology Discussion List - The primary functions of this list are: providing abstracts of papers posted to the Hopf archive, providing information about topology conferences, and serving as a forum for topics related to algebraic topology.
		Topology of Manifolds: Supersymmetry and QFT - This is the web resource page for a course taught by John Morgan in Fall 1997 at Columbia University.
		TTT on WWW - The Transpennine Topology Triangle is a topology seminar partially supported by the London Mathematical Society with vertices at Leicester, Manchester and Sheffield.
	dmoz.org /Science/Math/Topology (312 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.
		Most of the examples will be drawn on the 2-dimensional plane but, given the definitions of the distance and neighborhood could be carried over to the 1- and many dimensional cases.
	www.cut-the-knot.org /do_you_know/topology.shtml (759 words)

	Topology
		The logical topology is concerned with data transmission on the network,that is the methods used to pass information between workstations.
		All the communications travel along a common cable called a bus.A linear bus topology consists of a main run of a cable with a terminator at each end of the line.
		A Star topology is designed with each node (file server, workstations, and peripherals) connected directly to a central unit, the network hub or switch.
	www.albany.edu /~ap0349/isp523/web1/topology.html (595 words)

	General Topology - NoiseFactory Science Archives (http://noisefactory.co.uk)
		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.
		The topology on C is not an order topology - in fact, there is no possible ordering of the complex field which generates the standard topology.
	noisefactory.co.uk /maths/topology.html (4788 words)

	What is Topology? (Site not responding. Last check: 2007-10-22)
		Basically, topology is the modern version of geometry, the study of all different sorts of spaces.
		Topology is almost the most basic form of geometry there is. It is used in nearly all branches of mathematics in one form or another.
		We use topology to describe homotopy, but in homotopy theory we allow so many different transformations that the result is more like algebra than like topology.
	www.math.wayne.edu /~rrb/topology.html (567 words)

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