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Topic: Topological neighbourhood

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Continuous function

Limit (mathematics)

General topology

Topology

Behavior

Regular open set

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	Topology glossary
		A neighbourhood of a point p is a neighbourhood of the 1-point set {p}.
		A punctured neighbourhood of a point p is a neighbourhood of p, minus p.
		Topological spaces can be classified regarding the degree to which their points are separated, regarding their compactness, their overall size and their connectedness.
	www.ebroadcast.com.au /lookup/encyclopedia/lo/Local_base.html (1004 words)

	Plane (mathematics) - Wikipedia, the free encyclopedia
		The topological plane, or its equivalent the open disc, is the basic topological neighbourhood used to construct surfaces (or 2-manifolds) classified in low-dimensional topology.
		Isomorphisms of the topological plane are all continuous bijections.
		The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem.
	en.wikipedia.org /wiki/Plane_(mathematics) (1091 words)

	PlanetMath: homogeneous topological space
		This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous.
		This is true even if we do not require our manifolds to be paracompact, as any two points share a Euclidean neighbourhood, and a suitable homeomorphism for this neighbourhood can be extended to the whole manifold.
		This is version 2 of homogeneous topological space, born on 2006-10-07, modified 2006-10-14.
	planetmath.org /encyclopedia/HomogeneousTopologicalSpace.html (176 words)

	Regular space
		Most topological spaces studied in mathematical analysis are regular; in fact, they are usually completely regular, which is a stronger condition.
		Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G.
		Suppose that A is a set in a topological space X and f is a continuous function from A to a regular space Y.
	www.ebroadcast.com.au /lookup/encyclopedia/t3/T3_space.html (917 words)

	Kuratowski's Introduction to Topology
		On the other hand, the property of a curve to have a tangent line at every point is not a topological property; the circle has this property but the perimeter of a triangle does not, although it may be obtained from the circle by means of a homeomorphism.
		The generality of topological methods rests not only on the generality of the assumptions concerning the transformations considered but also on the generality of the sets considered to which these transformations are applied.
		This period was preceded by the transition from the investigation of subsets of Euclidean space in set-theoretic topology to the investigation of arbitrary topological spaces.
	www-groups.dcs.st-and.ac.uk /~history/Extras/Kuratowski_Topology.html (965 words)

	Topological space - Wikipedia, the free encyclopedia
		Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.
		Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset.
		Topological spaces provide the most common notions of closeness and convergence for a space, but it may be possible in some cases to study more specialized or more general notions.
	en.wikipedia.org /wiki/Topological_space (1934 words)

	Topological Foundations of Cognitive Science (Site not responding. Last check: 2007-10-03)
		Topological spatial properties will then in general fail to be invariant under more radical transformations, not only those which involve cutting or tearing, but also those which involve the gluing together of parts, or the drilling of holes through a body, or the decomposition of a body into separate constituent parts.
		Topological properties are discernible also in the temporal realm: they are those properties of temporal structures which are invariant under transformations of (for example) stretching (slowing down, speeding up) and temporal translocation.
		A preposition such as 'in' is magnitude neutral (in a thimble, in a volcano), shape neutral (in a well, in a trench), closure-neutral (in a bowl, in a ball); it is not however discontinuity neutral (in a bell-jar, in a bird cage).
	ontology.buffalo.edu /smith/articles/topo.html (5706 words)

	The GIS Primer - Mapping and Display
		Topological overlay is predominantly concerned with overlaying polygon data with polygon data, e.g.
		The result of a topological overlay in the vector domain is a new topological network that will contain attributes of the original input data layers.
		It is important to note that neighbourhood buffering is often categorized as being a proximity analysis capability.
	www.innovativegis.com /basis/primer/mapping.html (3345 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)

	Springer Online Reference Works
		A topological group is said to be connected, totally disconnected, compact, locally compact, etc., if the corresponding property holds for its underlying topological space.
		Every topological group is a uniform space in a natural way.
		The existence of a uniform structure on a topological group allows one to introduce and apply the notions of uniform continuity (for example, for real-valued functions on a topological group), Cauchy sequences, completeness, and completion.
	eom.springer.de /T/t093070.htm (1330 words)

	Neighbourhood
		The term neighbourhood is used for everything that is close, nearby, contiguous.
		The neighbourhood reaches its limit point (immediate neighbourhood) when the spatial units are contiguous, adjoining.
		The measure of topological relations may be used to express contiguities between objects or places which are close to each other, either because they have a common border if they are cells or zones, or because they are linked by a line if they correspond to nodes in a network.
	hypergeo.free.fr /article.php3?id_article=213 (321 words)

	Literature
		The methods that are used to prove these theorems are very similar to the techniques that are used to show that the fundamental group of a compact manifold with everywhere negative sectional curvature has exponential growth and has only cyclic abelian subgroups [6].
		The use of topological entropy is helpful, because of the following theorems that connect the Liouville-integrability of a geodesic flow on a compact manifold with zero topological entropy.
		The proof of the second and third conclusions relies on results due to Gromov [7, 8] and Yomdin [16] which place lower bounds on the entropy of a cogeodesic flow in terms of the rate of growth of the betti numbers of the loop space.
	www.mast.queensu.ca /~lbutler/thesisproposal/node4.html (1006 words)

	Oz's crib sheet (Site not responding. Last check: 2007-10-03)
		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)

	PlanetMath: boundary / frontier
		From the definition, it follows that the boundary of any set is a closed set.
		The term `boundary' (but not `frontier') is used in a different sense for topological manifolds: the boundary
		Cross-references: homeomorphic, neighbourhood, points, topological manifolds, closed set, closure, subset, topological space
	planetmath.org /encyclopedia/Boundary.html (123 words)

	[No title]
		The topological neighbourhood is defined as a function and will be used in the adaptation process (see section 2.1.3).
		To be specific, let EMBED Equation.3 denote the topological neighbourhood centered on winning neuron EMBED Equation.3 and encompassing a set of excited (cooperating) neurons, a typical on of which is denoted by EMBED Equation.3 .
		The algorithm therefore leads to a topological ordering of the feature map in the input space in the sense that neurons that are adjacent in the lattice will tend to have similar synaptic weight vectors.
	users.tkk.fi /~jblumme/som-harkka.doc (4976 words)

	Topology Course Lecture Notes
		The reader who has had abstract algebra will note that homeomorphism is the analogy in the setting of topological spaces and continuous functions to the notion of isomorphism in the setting of groups (or rings) and homomorphisms, and to that of linear isomorphism in the context of vector spaces and linear maps.
		A property of topological spaces which when possessed by a space is also possessed by every space homeomorphic to it is called a topological invariant.
		We learnt that, for metric spaces, sequential convergence was adequate to describe the topology of such spaces (in the sense that the basic primitives of `open set', `neighbourhood', `closure' etc. could be fully characterised in terms of sequential convergence).
	at.yorku.ca /i/a/a/b/23.dir/ch1.htm (2430 words)

	Springer Online Reference Works
		Absolute neighbourhood retracts are characterized as retracts of open subsets of convex subspaces of normed linear spaces.
		Conversely, two homotopy-equivalent spaces can always be imbedded in a third space in such a way that they are both deformation retracts of this space.
		"Absolute retract" and "absolute neighbourhood retract" are often abbreviated to AR and ANR.
	eom.springer.de /R/r081690.htm (561 words)

	Current Seminars
		ABSTRACT: We live in a topological space, and often information about some topology is used for reasons that are scientific or technical (eg., CAD or pattern recognition), or simply for entertainment.
		We will say that X is dually \bf P if any neighbourhood assignment in X has a kernel with the property \bf P.
		Then, from a graph we construct a particular compatible Alexandroff topological space said homeomorphic-equivalent to the graph.
	home.att.net /~topann/FebMay2006.html (865 words)

	The concentration phenomenon and topological groups by Vladimir Pestov
		Many `infinite-dimensional' topological groups G of importance have the fixed point on compacta property, or are extremely amenable, that is, every continuous action of G on a compact space has a fixed point.
		Every extremely amenable abelian topological group G is minimally almost periodic, that is, admits no non-trivial continuous characters.
		It is sensible to expect the concentration phenomenon growing in importance in topology when (and if) the `probabilistic' treatment of topological spaces, fucntions, and other objects becomes as widespread as the similar approach already is in geometric functional analysis and graph theory.
	at.yorku.ca /t/a/i/c/35.htm (1855 words)

	HOW THE PETRI-NETS ARE DERIVED FROM TREE-GENERATED ORDER . CONCEQUENCES TO THEIR IMPLEMENTATION . (Site not responding. Last check: 2007-10-03)
		A homotopy transformation of a path is a continuous transformation of it that fixes the end points and is realizable in a continuous way inside the topological space.
		One is purely algebraic and associates flow-charts to groups and makes use of the fact that any group is the quotient of a free group.
		That is we form the topological space T which is the disjoint union of X and Yx[0,1], where X and Y are provided with the discrete topology and the [0,1] has the topology of the real interval.Let R be the equivalence relation on T for which (y,t)= (
	www.softlab.ntua.gr /~kyritsis/PapersInComputerScience/Hrm98Stf.htm (1683 words)

	[No title] (Site not responding. Last check: 2007-10-03)
		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)

	Discrete Geometry for Computer Imagery (DGCI)
		Topological Quadrangulations of Closed Triangulated Surfaces Using the Reeb Graph
		Topological Operators on the Topological Graph of Frontiers
		A topologically consistent representation for image analysis: the Topological Graph of Frontiers
	wotan.liu.edu /docis/dbl/dgcidg/index.html (1589 words)

	CGSC5001: Cognition and Artificial Systems (Site not responding. Last check: 2007-10-03)
		You can adjust the number of output nodes in the Kohonen network by changing kNodes.
		You can turn off the 'neighbourhood' or 'topological map' aspect of the Kohonen network by removing the 'setNeighbourhoodFunc' command.
		Does it still perform as well when it is not a topological map?
	www.carleton.ca /ics/courses/cgsc5001/assign9.html (435 words)

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