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Topic: Closed (topology)

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	PlanetMath: Zariski topology (Site not responding. Last check: 2007-11-06)
		The Zariski topology is the predominant topology used in the study of algebraic geometry.
		Every regular morphism of varieties is continuous in the Zariski topology (but not every continuous map in the Zariski topology is a regular morphism).
		This is version 1 of Zariski topology, born on 2002-05-11.
	planetmath.org /encyclopedia/ZariskiTopology.html (150 words)

	Topology (Site not responding. Last check: 2007-11-06)
		A topology is a set of points, and a collection of subsets that are designated as "open".
		To be a proper topology, the empty set and the entire set must be open, and open sets remain open under finite intersection and arbitrary union.
		The complement of an open set is closed, and it follows that closed sets are closed under arbitrary intersection and finite union.
	www.mathreference.com /top,intro.html (430 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)

	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)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		Topological structure (topology)); it is immaterial whether this is an open or closed topology (one transfers into the other by replacing the sets constituting the given topology by their complements).
		to specify a base of the given topology), in terms of which all remaining elements of the topology can be obtained as unions (in the case of an open topology) or intersections (in the closed case) of sets belonging to the base.
		Closely associated with the concept of a topological space is that of a continuous mapping from one space into another.
	eom.springer.de /T/t093130.htm (2209 words)

	Topology history
		The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common.
		Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler.
		A second way in which topology developed was through the generalisation of the ideas of convergence.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Topology_in_mathematics.html (1456 words)

	PlanetMath: closed set
		Closed subsets can also be characterized as follows:
		is not closed under the standard topology on
		This is version 4 of closed set, born on 2002-03-02, modified 2006-10-22.
	planetmath.org /encyclopedia/Closed.html (76 words)

	Closed set - Wikipedia, the free encyclopedia
		In topology and related branches of mathematics, a closed set is a set whose complement is open.
		The unit interval [0,1] is closed in the real numbers, and the set [0,1] ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q is not closed in the real numbers.
		The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces.
	en.wikipedia.org /wiki/Closed_set (586 words)

	Cartan's Corner : Point Set Topology
		Remarks: In the definition of a topology when the number of elements of the set in not finite, the logical intersection of open sets is restricted to any pair, and the logical union of closed sets to restricted to any pair.
		The closure of (ab) relative to the topology T4(open) is
		Relative to the topology T4(open), the interior of (ab) is the singleton, (a):
	www22.pair.com /csdc/car/carfre64.htm (2727 words)

	PlanetMath: product topology
		The product topology is generally more useful than the box topology.
		The main reason for this can be expressed in terms of category theory: the product topology is the topology of the direct categorical product in the category Top (see Theorem 1 below).
		This is version 30 of product topology, born on 2002-06-12, modified 2006-10-01.
	planetmath.org /encyclopedia/ProductTopology.html (182 words)

	Topology glossary - Wikipedia, the free encyclopedia
		Each connected component is closed, and the set of connected components of a space is a partition of that space.
		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.
	en.wikipedia.org /wiki/Topology_glossary (4707 words)

	Closed extension topology - Wikipedia, the free encyclopedia
		The Closed extension topology is a topology created on the extension of a set by extending the closed sets.
		We note the closed sets on X\{p} are the same as those on X, which shows the equivalence of the two definitions.
		If we replace the criteria that X\S be closed with the criteria that it be compact then we converted from the closed extension topology to the one-point compactification.
	en.wikipedia.org /wiki/Closed_extension_topology (260 words)

	Closed graph theorem - Wikipedia, the free encyclopedia
		In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.
		The restriction on the domain is needed due to the existence of closed unbounded linear operators.
		The usual proof of the closed graph theorem employs the open mapping theorem.
	en.wikipedia.org /wiki/Closed_graph_theorem (268 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.
		Hence, if A is any nonempty subset of X, the intersection of all closed subsets of X which contain A must also be closed, and we call this set the _closure_ of A, denoted cl(A), or cl(A,t) if it is necessary to indicate the topology t on X with respect to which closure is taken.
	www.georgetown.edu /faculty/kainen/topology.html (1132 words)

	Topology Course Lecture Notes
		We all recall the important and useful theorem from calculus, that functions which are continuous on a closed and bounded interval take on a maximum and minimum value on that interval.
		This topology is 'just right' in the sense that it is barely fine enough to guarantee the continuity of the coordinate projection functions while being just course enough allow the important result of Theorem.
		A basic formal distinction between algebra and topology is that although the inverse of a one-one, onto group homomorphism [etc!] is automatically a homomorphism again, the inverse of a one-one, onto continuous map can fail to be continuous.
	at.yorku.ca /i/a/a/b/23.dir/index.htm (8277 words)

	Topological Preliminaries
		Topology is one of (quite a few) mathematical theories that permeate other branches of Mathematics connecting them into one coherent whole.
		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.
		b} is closed in R and dense in itself.
	www.cut-the-knot.org /do_you_know/topology.shtml (759 words)

	University of Chicago Algebraic Topology Seminar
		The algebraic topology seminar is held in Eckhart Hall room 203, on Tuesdays at 4:30PM, unless otherwise specified.
		The area of string topology began with a construction by Chas and Sullivan of previously undiscovered algebraic structure on the homology H_(LM) of the free loop space of an oriented manifold M. Among other results, Chas and Sullivan showed that H_(LM), suitably regraded, carries the structure of a graded-commutative algebra.
		Open-closed string topology, first sketched by Sullivan, arises when considering spaces of paths in M with endpoints constrained to lie on given submanifolds (the so-called D-branes).
	www.math.uchicago.edu /~jg/winter05topsem.html (1243 words)

	Banach space preliminaries
		The weak topology on X is the weakest topology such that every bounded linear functional on X is continuous.
		In particular, the weak-* topology is not in general metrizable, even when X is separable.
		is a closed subset of a complete metric space, written as a countable union of closed sets.
	www.math.psu.edu /simpson/papers/convex-l/node2.html (999 words)

	Computational Topology for Regular Closed Sets (within the I-TANGO project) by T.J. Peters et al (Site not responding. Last check: 2007-11-06)
		The Boolean algebra of regular closed sets is prominent in topology, particularly as a dual for the Stone-Cech compactification.
		The acronym I-TANGO is an abbreviation for "Intersections - Topology, Accuracy and Numerics for Geometric Objects".
		Distributed by Topology Atlas with permission of the author.
	at.yorku.ca /t/a/i/c/50.htm (175 words)

	Glossary of Topology Terms
		a space A is an absolute retract if whenever a normal space X has a closed subspace B homeomorphic to A, then B is a retract of X.
		The metric d gives rise to a topology on P; a basis for this topology is the collection of sets { q: d(p,q)0.
		In other words, a subset S of P is open in the metric topology if for every point p in S there is a number e>0 such that S contains the set { q: d(p,q)
	cage.rug.ac.be /~hvernaev/TopoGloss.html (2536 words)

	Cosmology :: Physics (Site not responding. Last check: 2007-11-06)
		Cosmic Topology - General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi- rather than simply- connected.
		This paper reviews the mathematical properties of multi-connected spaces, and the different tools to classify them and to analyse their properties.
		Topology of the Universe - A thesis (in PDF format) which studies the possible topologies of the universe and their physical implications.
	science.gourt.com /Physics/Cosmology.html (1597 words)

	Draw star of pentagram, pentacle, pentagon: instruction & illustration
		Because the electron's momentum is also a wave (de Broglie), the electron's wave must evenly, that is exactly, close upon itself to form a standing and a round wave that is symmetrical about a point.
		The mainstream scientists' argument that "computer's representation of an irrational number is close enough" is, unfortunately, not relevant to atomic construction.
		Proving or disproving one task is good for both topologies, for either topology is commensurable (proportional) to the other through a real constant that is radius.
	www.hyperflight.com /pentagon-construct.htm (5671 words)

	Advanced Data Path Routing Techniques for Three Tier KVM Switch Networks - Page 8 of 9
		Or, the number of system chassis user ports (outputs); when chassis' are installed in the same racks as the servers.
		In a closed topology, Router and User chassis will be installed in the same rack and don't require patch panels.
		Using a standard identification and connection pattern between router chassis and the user/system chassis makes installation easy.
	www.tron.com /dpr00008.html (221 words)

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