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Topic: Topological interior

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	[No title]
		Technically, if X is a topological space with a fiber bundle F, it is the minimum number of open sets that cover X such that F is trivial over each open set.
		The homology of a topological space has a relatively technical definition, but it is relatively easy to compute and study with tools from linear algebra.
		loop space Given a topological space X, its loop space is the topological space of all continuous functions from a circle to X. Loop spaces are important examples of new topological spaces formed from old ones, as well as examples of infinite-dimensional spaces in mathematics.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	Dynamic Topological Logic
		Dynamic Topological Logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic.
		Topological dynamics studies the asymptotic properties of continuous maps on topological spaces.
		Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality □ (interior), and two temporal modalities, ○ (a circle for "next") and * (an asterisk for "henceforth"), both interpreted using the continuous function f.
	individual.utoronto.ca /philipkremer/DynamicTopologicalLogic.html (649 words)

	A 3D spatial data model for terrain reasoning
		However, a node may be located within the interior of a face or within the interior of a volume.
		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.
		Several of the topological relationships defined in the preceding sections depend on the level of topology that is present.
	www.geovista.psu.edu /sites/geocomp99/Gc99/037/gc_037.htm (4908 words)

	Good Math, Bad Math : Shapes, Boundaries, and Interiors
		In a topological space, the basic notion of distance is built on neighborhoods.
		The interior of {b,c} is {b,c}, {a,b}*={a,b,c}, and the boundary of {a,b} is empty.
		A point in a topological space is just any object that's a member of a family of sets that has the correct properties to define a topology.
	scienceblogs.com /goodmath/2006/09/shapes_boundaries_and_interior.php (3162 words)

	Topological mapping with sensing-limited robots (Site not responding. Last check: 2007-10-10)
		Topological maps represent the connectivity of the environment, usually in a graph structure where vertices are "distinctive places" in the environment and edges represent paths, classes of paths or behavior sequences for traveling between places.
		Though not always, many topological maps are augmented with some metric information, such as the lengths of paths (based on the robot's odometry).
		Others have employed the basic GVD as a topological map, with meet points in the GVD as nodes in the map, and edges in the GVD as edges in the map.
	www.cs.rpi.edu /~beevek/mapping.html (1003 words)

	PlanetMath: interior
		The interior of a set enjoys many special properties, some of which are listed below:
		This is version 11 of interior, born on 2002-06-21, modified 2006-09-09.
		Object id is 3123, canonical name is Interior.
	planetmath.org /encyclopedia/Interior.html (82 words)

	Amazon.com: "topological interior": Key Phrase page (Site not responding. Last check: 2007-10-10)
		However, if we define the topological interior of a set H of points in R3 as the set of all those points v, for which there exists...
		and the interior and boundary of a as a simplex may not be equal to its topological interior and boundary as a subset of R".
		The (topological) interior of any set SZ, denoted SZ, is the set of points in SZ which are the centers of some balls...
	g.msn.com /9SE/1?http://amazon.com/phrase/topological-interior&&DI=6244&IG=a8a69406f5634276a39083bf64b78ef9&POS=3&CM=WPU&CE=3&CS=AWP&SR=3 (520 words)

	Topological condensate concept
		One ends up with the concept of the topological condensate from the requirement that both TGD as a Poincare invariant gravity and TGD as generalization of the string model pictures make sense as appropriate limiting cases.
		Topological condensate has a hierarchical structure and one can associate a characteristic length scale spesifying the minimum size of the 3-surfaces on each level of the hierarchy.
		This picture of topological condensate is consistent with the general properties of the Kähler action.
	www.physics.helsinki.fi /~matpitka/topcond.html (487 words)

	Algebraic Topology: Topology
		A topological space is a set X together with a collection of subsets OS the members of which are called open, with the property that (i) the union of an arbitrary collection of open sets is open, and (ii) the intersection of a finite collection of open sets is open.
		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.
		A topological space is said to be locally P for some property P when for each point x and neighbourhood U of x there is a set A contained in U and containing a neighbourhood of x that has property P.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	Open Shortest Path First (OSPF)
		Similar to the Interior Gateway Routing Protocol (IGRP), OSPF was created because in the mid-1980s, the Routing Information Protocol (RIP) was increasingly incapable of serving large, heterogeneous internetworks.
		A topological database is essentially an overall picture of networks in relationship to routers.
		Topological databases are synchronized between pairs of adjacent routers.
	www.cisco.com /univercd/cc/td/doc/cisintwk/ito_doc/ospf.htm (1581 words)

	Errata and Addenda for CNL
		Perhaps a better order for this material would be to the definition of "interior" from 4.10 to immediately after 4.5, so that each of the examples would only have to be covered once.
		The interior of a set then turns out to be equal to the union of all the open intervals contained in that set.
		as a subset of a topological space, where the topological space is R with one of the three topologies listed.
	www.math.vanderbilt.edu /~schectex/logics/errata/errata.html (1098 words)

	Topological Properties
		These differences are not important for basic topological properties, so statements and proofs involving H are often identical to those for R. First an open ball of quaternions needs to be defined to set the stage for an open set.
		The interior of A is the union of all open sets contained within A. The interior equals A if and only if A is open.
		A continuous function from a compact topological space into H is bound and its absolute value attains a maximum and minimum values.
	www.theworld.com /~sweetser/quaternions/intro/topology/topology.html (1623 words)

	Topology MAT 530
		The next were the formal definitions of metric and topological spaces, bases and subbases in topological spaces (i.e., a description of different ways to define topology on a set).
		We gave a topological definition of continuity that does not appeal to a metric structure or any other additional structures and thus can be applied to any topological spaces.
		The Urysohn lemma states that for a normal topological space X and two disjoint closed subsets A and B of it, there exists a continuous function from X to [0,1] that is 0 on A and 1 on B.
	www.math.sunysb.edu /~timorin/mat530.html (2896 words)

	Topological Spatial Relations (Site not responding. Last check: 2007-10-10)
		It corresponds to the topological type of a pair of intersecting sets in a topological space.
		The previously introduced 4-intersection (boundary-boundary, interior-interior, boundary-interior, and interior-interior), a topological invariant of topological spatial relations, is further refined by considering the number of components and the dimension of each component in each intersection in the 4-intersection.
		We show how the topological spatial relations between two 2-disks in the plane can be classified using these invariants.
	www.spatial.maine.edu /~max/CP22.html (91 words)

	Oz's crib sheet (Site not responding. Last check: 2007-10-10)
		A topological space can also be defined in terms of closed sets, interior operators, neighbourhoods, and other different but equivalent ways.
		Topological property: a spatial property that depends purely on continuity, and not on differentiability, or distance, or parallelism, or angles, or any of that stuff we visualize so readily.
		Examples of topological properties: «closed curve C in the plane doesn't intersect itself», «point p is inside closed curve C that doesn't intersect itself».
	math.ucr.edu /~toby/Oz (579 words)

	[No title]
		Topological relations are captured by boundary and co-boundary relations which connect adjacent cells.
		Topological models have been described by Corbett (1979), White (1979), Frank and Kuhn (1986), and implementation aspects by Jackson (1989) and Egenhofer et al.
		The purpose of topological representations is to model the topological properties of a selected part of the world.
	www.ncgia.ucsb.edu /Publications/Tech_Reports/91/91-17.doc (1877 words)

	That Logic Blog: November 2005
		Pick your favourite topological space X and let O(X) be the collection of all open sets of X. Then, O(X) is a complete Heyting algebra, with join defined by union and meet almost defined by set intersection.
		Now, O(X) directly forms a model of Int by interpreting conjunction as the interior of intersection, disjunction as union, implication as the subset relation and negation as the interior of the relative complement.
		This is precisely what we need, since an interior operator uniquely determines a topology of open sets (this is usually stated in terms of the dual notion of a closure operator and holds for a large variety of lattices, not just topological spaces).
	thatlogicblog.blogspot.com /2005_11_01_archive.html (1644 words)

	Topological Curiosities
		Thus squeezing and shrinking are also topological transformations while gluing, having tearing as the reverse transformation, is not topological.
		Topology includes the study of knots, one-sided surfaces and dimensionality of curves and surfaces among a variety of other things.
		Far from dealing with obvious, the topological research discovered things even more marvelous than the plane filling curves.
	www.cut-the-knot.org /do_you_know/brouwer.shtml (455 words)

	[No title]
		There, he interpreted the generation of the support D of the image from a topological space S by means of some 'discretization' as the construction of a quotient space $Delta of S, which represents the set D an d has a reasonable (non- discrete) topology.
		The previously introduced 4- intersection (boundary-boundary, interior-interior, boundary- interior, and interior-interior), a topological invariant of topological spatial relations, is further refined by considering the number of components and the dimension of each component in each intersection in the 4-intersection.
		We show how the topological spatial relations between two 2-disks in the plane can be classified using these invariants.!6 Relationship between two views of a 3D object scene, the camera (displacement) model, and the scene structure, pp.247-255 Author(s): John Merchant, Loral Infrared and Imaging Systems, Lexington, MA, USA.
	www.spie.org /web/abstracts/1800/1832.html (4058 words)

	Galois Correspondence (Site not responding. Last check: 2007-10-10)
		Let us write o for interior and - for closure, so that the closure of the interior of A is written Ao-.
		A subset A of a topological space X is called regular open when A = A-o.
		Example In a topological space X take for both P and Q the power set of X ordered by inclusion, and take f = g = -c where c is taking complements and - is the closure operator.
	www.win.tue.nl /~aeb/at/GaloisCorrespondence.html (334 words)

	3D Boolean Operations on Nef Polyhedra (Site not responding. Last check: 2007-10-10)
		The boolean operations are not evaluated, instead, objects are represented implicitly with a tree structure; leaves represent primitive objects and interior nodes represent boolean operations or rigid motions, e.g., translation and rotation.
		Surfaces are oriented to decide between the interior and exterior of a solid.
		Set complement changes between open and closed halfspaces, thus the topological operations boundary, interior, exterior, closure and regularization are also in the modeling space of Nef polyhedra.
	www.cgal.org /Manual/3.2/doc_html/cgal_manual/Nef_3/Chapter_main.html (3571 words)

	Amazon.com: "interior operator": Key Phrase page (Site not responding. Last check: 2007-10-10)
		Then int is the interior operator for a topology on X if and only if int satisfies these four conditions: int(X) = X, S D int(S),...
		and sometimes by the use of an interior operator I. In the former case, a topological space is a set S and a closure mapping K on S that...
		Affordable Interior Designer in Spokane -- Sheik Designs is dedicated to make your interior design dreams come to life.
	g.msn.com /9SE/1?http://amazon.com/phrase/interior-operator&&DI=6244&IG=a8a69406f5634276a39083bf64b78ef9&POS=4&CM=WPU&CE=4&CS=AWP&SR=4 (582 words)

	Selection in Context: Patterns of Natural Selection in the Glycoprotein 120 Region of Human Immunodeficiency Virus 1 ...
		Hence, interior haplotypes have demonstrated a degree of evolutionary
		These lines are solid if the branch is an intravisit interior, dashed if an intravisit tip, and dotted if an intervisit branch.
		Vertical lines do not indicate any evolutionary change, but rather are used to show when multiple lineages diverge from a single ancestral haplotype or node in the tree.
	www.genetics.org /cgi/content/full/167/4/1547 (7999 words)

	Computational Geometry Package: Simplicial Mesh Package
		In 3-D, local topological transformations replace a set of tetrahedra with a different set that fills the same domain.
		We have been implementing the algorithms presented in ``Two Discrete Optimization Algorithms for the Topological Improvement of Tetrahedral Meshes'' by Jonathan Shewchuk.
		A better approach is to keep track of the edges and faces upon which local topological changes could possibly improve the mesh.
	www.cacr.caltech.edu /~sean/projects/stlib/html/geom/simplicial.html (437 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Topo-bisimulations: topological analogue of modal bisimulation: somewhat coarse zigzag relations between topological spaces with a valuation, that encode winning strategies for Duplicator.
		Bisimulation contractions of topological region patterns can be ordinary Kripke models, which then serve as discrete caricatures of the continuous visual situation.
		Example: determining logics of reasonable sets that occur in 'images' (a), or are needed in competeness proofs (b).
	www.stanford.edu /~sarenac/Stanford_Space.doc (1249 words)

	About Fractals (Site not responding. Last check: 2007-10-10)
		A: Topological dimension is the "normal" idea of dimension; a point has topological dimension 0, a line has topological dimension 1, a surface has topological dimension 2, etc.
		A set S has topological dimension k if each point in S has arbitrarily small neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the least nonnegative integer for which this holds.
		A strange attractor is an attractor that is topologically distinct from a periodic orbit or a limit cycle.
	avatargraphics.com /fractalland/fractalfaq.html (6025 words)

	3D Nef Polyhedron (Site not responding. Last check: 2007-10-10)
		Starting from halfspaces (and also directly from oriented 2-manifolds), we can work with set union, set intersection, set difference, set complement, interior, exterior, boundary, closure, and regularization operations (see Section
		Set union, difference and symmetric difference can be reduced to intersection and complement.
		The first volume is the outer volume and the second volume is the interior of the cube.
	www.ics.uci.edu /~dock/manuals/cgal_manual/Nef_3/Chapter_main.html (3353 words)

	Fractal Frequently Asked Questions and Answers
		Benoit Mandelbrot gives a mathematical definition of a fractal as a set for which the Hausdorff Besicovich dimension strictly exceeds the topological dimension.
		A4b: Topological dimension is the "normal" idea of dimension; a point has topological dimension 0, a line has topological dimension 1, a surface has topological dimension 2, etc. For a rigorous definition: A set has topological dimension 0 if every point has arbitrarily small neighborhoods whose boundaries do not intersect the set.
		(Since the boundary has empty interior, the topological dimension is less than 2, and thus is 1.) Reference: 1.
	www.faqs.org /faqs/fractal-faq (9659 words)

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