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# Topic: Interior (topology)

 Interior -- from Wolfram MathWorld The interior of a set is the union of all its open subsets. More informally, the interior of geometric structure is that portion of a region lying "inside" a specified boundary. ball and the interior of a circle is an (open) disk. mathworld.wolfram.com /Interior.html   (71 words)

 Interior (topology) - Wikipedia, the free encyclopedia In mathematics, the interior of a set S consists of all points which are intuitively "not on the edge of S". The notion of interior is in many ways dual to the notion of closure. The interior of a set S is the set of all interior points of S. en.wikipedia.org /wiki/Interior   (660 words)

 Alexandrov topology - Wikipedia, the free encyclopedia Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of X, this construction is a special case of the construction of a modal algebra from a modal frame i.e. With the advancement of categorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of finite generation was applied to general topology and the name finitely generated spaces was adopted for them. Inspired by the use of Alexandrov topologies in computer science, applied mathematicians and physicists in the late 1990's began investigating the Alexandrov topology corresponding to causal sets which arise from a preorder defined on spacetime modeling causality. en.wikipedia.org /wiki/Alexandrov_topology   (1355 words)

 Boundary (topology) - Wikipedia, the free encyclopedia In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. One should keep in mind the boundary of a set is a topological notion, therefore, one changes the topology, the set boundary may change. en.wikipedia.org /wiki/Boundary_(topology)   (630 words)

 Topology glossary : Topological interior 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. The interior of a set is the union of all open sets contained in it. www.fastload.org /to/Topological_interior.html   (1051 words)

 [No title]   (Site not responding. Last check: 2007-10-26) As another example, a bucket field topology having relatively large lobes can be imposed on the plasma during the initial stages of processing, then a bucket field topology having intermediate sized lobes can be imposed on the plasma, and then a bucket field topology having relatively small lobes can be imposed on the plasma. The method of claim 3, wherein the at least one rotating magnetic field topology corrects a non-uniformity in plasma density of said plasma while said at least one rotating magnetic field topology is imposed on said plasma. The method of claim 9, wherein a plurality of bucket field topologies are imposed on the plasma to decrease plasma density at a predetermined rate during the first portion of the plasma processing operation. www.wipo.int /cgi-pct/guest/getbykey5?KEY=03/25971.030327&ELEMENT_SET=DECL   (6778 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)

 Cartan's Corner : Point Set Topology 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): In all cases, note that the union of the interior and the boundary is equal to the union of the set and its limit points. www22.pair.com /csdc/car/carfre64.htm   (2727 words)

 A 3D spatial data model for terrain reasoning It is possible for a node to be in the interior of a face, while also being the endpoint of an edge which does not bound that face, as shown in Figure 1. However, in 3D topology, the collection of edges that are associated with a node cannot always be completely ordered, as the edges may connect to the node from any direction in three-dimensional space. The 2D topology model also allows for edges, or collections of connected edges, which are "floating" within a face (i.e., not attached to the outer ring of the face). www.geocomputation.org /1999/037/gc_037.htm   (4908 words)

 Topological Properties T is the topology on H. The subsets of H in T are open sets. Quaternions form a topology because they are what mathematicians call a metric space, since q* q evaluates to a real positive number or equals zero only if q is zero. Topology plus causality could be the key for subdividing different regions of physics. world.std.com /~sweetser/quaternions/intro/topology/topology.html   (1623 words)

 Topological Foundations of Cognitive Science Topology in this mathematical sense has been used by cognitive scientists in work on the mathematical properties of connectionist networks and elsewhere. Of all the precursors of contemporary applications of topology in cognitive science, the most notorious is the work on "topological and vector psychology" of the German Gestalt psychologist Kurt Lewin. Topology can serve as a theoretical basis for a unification of diverse types of psychological facts. ontology.buffalo.edu /smith/articles/topo.html   (5706 words)

 Baire space: Encyclopedia topic   (Site not responding. Last check: 2007-10-26) The property of being a Baire space is a topological property (topological property: in topology and related areas of mathematics a topological property or topological... The interior of every union (union: The state of being joined or united or linked) of countably many nowhere dense (nowhere dense: more facts about this subject) sets is empty set. As a topological space, B is homeomorphic (homeomorphic: in the mathematical field of topology a homeomorphism or topological isomorphism... www.absoluteastronomy.com /reference/baire_space   (1715 words)

 PlanetMath: CW complex Remark 1   There is potential for confusion in the way words like “open” and “interior” are used for cell complexes. it is not necessarily the case that it is the interior of Cross-references: open, interior, open set, potential, theory, category, loop spaces, infinite-dimensional, varieties, algebraic, manifolds, finite-dimensional, smooth, homeomorphic, even, topological spaces, simplicial complexes, homotopy equivalent, homotopy, dimensions, order, cell attachment, discrete space, structure, attaching maps, subspace topology, intersection, closed, cells, weak topology, open cells, union, finite, closed cell, collection, subspaces, filtration, complex, Hausdorff topological space planetmath.org /encyclopedia/CWComplex2.html   (443 words)

 [No title] It goes like this: Let the space be the natural numbers with the topology consisting of all sets of the form {1, 2,..., n} plus the empty set and the whole space. Rabinowitz seemed to have a knack for picking just the right topology and right seed set to generate the power set of the space under the three operations; he was able to do this for every cardinality of the underlying space we tried. N. Levine, On the commutativity of the closure and interior operators in topological spaces, {\it ibid.}, {\bf 68} (1961) 474-477. www.math.niu.edu /~rusin/known-math/99/kuratowski   (1339 words)

 TCP/IP Tutorial and Technical Overview Interior routing protocols or interior gateway protocols (IGPs) are used to exchange routing information between routers within a single autonomous system. The topology of an area is represented with a database called a Link State Database describing all of the links that each of the routers in the area has. Routers are divided into Level 1 routers, which know nothing of the topology outside their areas, and Level 2 routers, which do know about the higher-level topology, but know nothing about the topology inside the areas unless they are also Level 1 routers. www.cas.mcmaster.ca /~cs4cd3/tcpip/3376c33.htm   (10640 words)

 Papier   (Site not responding. Last check: 2007-10-26) The notions of Granule, Topology and Set_relationship are defined to guarantee a semantics to alphanumeric data parts associated to the results of spatial database queries. The spatial representation of a geographical object is defined by a boundary and an interior (Egenhoffer 1989). The Topology defines the validity on the boundary, the interior or the entirely spatial representation (i.e., Global stands for boundary and interior) for a given attribute. www-inf.int-evry.fr /~adm_bd/Cigales/Publications/Cig_Publi_html/CEUS94.html   (3717 words)

 Algebraic Topology: Topology 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. A topological space is called metric when there is a distance function determining the topology (i.e., open balls for the metric are open sets, and conversely, if a point x lies in an open set U then for some positive e the ball with radius e around x is contained in U. www.win.tue.nl /~aeb/at/algtop-2.html   (1509 words)

 Exercises 5 Show that the subspace topology on any finite subset of R is the discrete topology. Show that the subspace topology on the subset Z is not discrete. Show that there are 29 different topologies on the set {a, b, c}. www-groups.dcs.st-and.ac.uk /~john/MT4522/Tutorials/T5.html   (187 words)

 [No title]   (Site not responding. Last check: 2007-10-26) end: # Next function computes the topology generated by a family F # of subsets of a given set X. It works by closing the family # respect to intersection and union. topology := (X,F) -> close_family(close_family(F,intersect,X),union,X): # Next function decides if a family F of subsets of a set X is # a topology. interior := proc(S,X,T) local A,R: if not type(T,set) then ERROR(Third argument,T,should be a set.) fi: R := {}: for A in T do if A intersect S = A then R := R union A fi: od: R: end: # Next function computes the closure of a set S in a topological # space (X,T). www.math.northwestern.edu /~mlerma/software/topology   (550 words)

 Interior (topology)   (Site not responding. Last check: 2007-10-26) A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. The interior of a set S is the set of all interior points of S. www.pillscatalog.net /Interior_(topology).html   (784 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)

 ISP Column - Exploring AS Numbers These network domains use an interior routing protocol (commonly referred to as an “IGP”, after the term "interior gateway protocol”) which maintains a complete mapping set of the current internal topology of the domain, together with the set of “best paths” between any two points within the network domain. To partition the extent to which this fine-grained topology information is propagated across the Internet, the approach used within the Internet’s routing architecture is to call upon a second level of routing hierarchy. In the inter-domain space a path to an address is described as a sequence of domains that must be transited to reach the domain that originates the particular address prefix. ispcolumn.isoc.org /2005-08/as1.html   (2550 words)

 Topology/Metric Spaces - Wikibooks, collection of open-content textbooks The interior of a set A is the Set of all the internal points of A. (do not confuse with the topological definition of the interior) its metric topology is the topology induced by using the set of all open balls as the base. One can also define the topology induced by the metric, as the set of all open groups defined by the metric. en.wikibooks.org /wiki/Metric_Spaces   (315 words)

 The second strategy:   (Site not responding. Last check: 2007-10-26) This reading of connection, while allowing two abutting objects to be externally connected, leads to a very unpalatable consequence: that the interior of a region is connected to the exterior. Since there is overlap between the closure of the exterior and the closure of the interior, they are connected. Furthermore, since the closure of a region is connected to the same stuff that the interior of the region is connected to, the closure of a region ends up being a part of the interior of the region. www.columbia.edu /~cbh2002/mt/2.htm   (763 words)

 Carfree Cities: Blocks However, no reasonable topology and transport scheme can keep maximum transport times within the metropolitan area well under one hour unless densities are high. The use of the interior courtyards (the open space inside a city block) is sure to give rise to debate. Interior courtyards admit daylight to building interiors and provide green space adjacent to virtually every building. www.carfree.com /block.html   (429 words)

 ESMN research   (Site not responding. Last check: 2007-10-26) The ESMN studies the structure and dynamics of solar surface fields, the topology and evolution of solar active regions, and the electrodynamical coupling between the solar interior, photosphere, and outer atmosphere by perfecting solar magnetometry, organising joint multi-telescope observing campaigns, and analysing the results through numerical inversions and simulations. The ESMN goal is to gain basic insight in the roots of solar magnetism by establishing the structure and dynamics of magnetic fields at the solar surface, charting the patterns that constrain the solar dynamo, and identifying the magnetic coupling between the different solar regimes from the interior to the corona. The precursor geometry may be identified through comparison of the before-and-after coronal field topology derived from high-resolution surface magnetometry, with extrapolative identification of magnetic nulls, separators, separatrices and quasi-separatrix layers where reconnection may occur, and with special emphasis on the topological role of field helicity. esmn.astro.uu.nl /ESMN_research.html   (2500 words)

 Interior   (Site not responding. Last check: 2007-10-26) In general, the interior of something refers to the space or part inside of it, excluding any kind of wall or boundary around its outside. For the set function used in topology, see interior (topology). This is a disambiguation page; that is, one that just points to other pages that might otherwise have the same name. bopedia.com /en/wikipedia/i/in/interior.html   (102 words)

 Interior Point Multigrid Methods For Topology Optimization - Maar, Schulz (ResearchIndex)   (Site not responding. Last check: 2007-10-26) In this paper, a new multigrid interior point approach to topology optimization problems in the context of the homogenization method is presented. The key observation is that nonlinear interior point methods lead to linear-quadratic subproblems with structures that can be favorably exploited within multigrid methods. 15 the local behavior of an interior point method for nonlinear.. citeseer.ist.psu.edu /59475.html   (615 words)

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