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Topic: Topological properties

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	PlanetMath: classification of topological properties according to behaviour under mapping
		Topological properties may be classified by their behaviour with respect to mappings.
		A property of a topological space is called continuous if it is the case that, whenever a space has this property, the images of this space under all continuous mapping also have the same property.
		This is version 11 of classification of topological properties according to behaviour under mapping, born on 2004-09-24, modified 2006-12-21.
	planetmath.org /encyclopedia/Continuous3.html (407 words)

	Springer Online Reference Works
		The consideration of topological properties of spaces of mappings is useful in proving theorems on the existence of mappings with some property.
		The fact that duality between properties of a topological space and topological properties of the space of functions on them with the topology of pointwise convergence is inherited is of special significance.
		The quest for categories of topological spaces with nice categorical properties, such as Cartesian closedness, has also led to other ways of capturing the idea of "nearness" (in analysis, geometry and elsewhere) and has contributed to the emergence of categorical topology and the theory of topological structures, [a5].
	eom.springer.de /s/s086200.htm (966 words)

	Topological Foundations of Cognitive Science
		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.
		The property of being a (single, connected) body is a topological spatial property, as also are certain properties relating to the possession of holes (more specifically: properties relating to the possession of tunnels and internal cavities).
		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.
	ontology.buffalo.edu /smith/articles/topo.html (5706 words)

	Topology - Wikipedia, the free encyclopedia
		In current usage, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.
		A toroid in three dimensions; A coffee cup with a handle and a donut are both topologically indistinguishable from this toroid.
		A traditional joke is that a topologist can't tell the coffee mug out of which she is drinking from the doughnut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
	en.wikipedia.org /wiki/Topology (1823 words)

	PlanetMath: index of properties of topological spaces
		The puropse of this meta-entry is to list all topological properties discussed in this encyclopaedia (or soon to be added).
		"index of properties of topological spaces" is owned by rspuzio.
		This is version 21 of index of properties of topological spaces, born on 2004-09-29, modified 2006-10-13.
	planetmath.org /encyclopedia/IndexOfProperiesOfTopolologicalSpaces.html (120 words)

	Good Math, Bad Math : Topological Spaces
		The properties we worked out for the open and closed sets are exactly the properties that are required of the open and closed sets of the topology.
		Topological spaces are basically sets with extra structure, and morphisms in that category are simply maps of sets preserving that extra structure, and since injectivity and surjectivity are set theoretic concepts, nothing changes in that respect from Top to Sets.
		I think your explanation of the relationship between the formal topological notion of continuity and the intuitive notion of a continuous function leaves something to be desired.
	scienceblogs.com /goodmath/2006/08/topological_spaces.php (1438 words)

	science
		Understanding of elementary topological equivalencies is impaired by preconceptions about the topological structure of ordinary objects, so that the equivalencies turn out to be counterintuitive.
		In spite of this an many other "bizarre" topological facts being widely known (their analysis often constitutes a standard exercise in topology textbooks) an explanation of why they appear bizarre and counterintuitive is still forthcoming.
		An explanation of the incorrect analysis of the topological equivalencies including double toruses might make appeal to different factors, but a simple solution is available when one construes the cognitive representation of holed objects as making ineliminable reference to holes.
	roberto.casati.free.fr /shadowmill.com/naivetopo.htm (2189 words)

	James Tauber : Poincare Project: Connectedness, Closed Sets and Topological Properties
		Topological spaces for which this is not true are said to be connected.
		A topological space is connected if and only if the only two sets that are both open and closed are the empty set and the set of all points.
		Because topological properties are based only on the open sets, they are preserved by a homeomorphism.
	jtauber.com /blog/2004/12/07/poincare_project:_connectedness,_closed_sets_and_topological_properties (497 words)

	Topological manifold - Wikipedia, the free encyclopedia
		Differentiable manifolds, for example, are topological manifolds equipped with a differential structure.
		Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold.
		The dimension of a manifold is a topological property, meaning that any manifold homeomorphic to an n-manifold also has dimension n.
	en.wikipedia.org /wiki/Topological_manifold (1446 words)

	Topological Properties (Site not responding. Last check: 2007-11-03)
		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.
		This property is used to analyze compactness, something vital for rigorously establishing differentiation and integration.
		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)

	Oz's crib sheet
		When speaking of it as a topological space, you are forbidden to talk about any of the non-topological properties of the space.
		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)

	Research in the Topological Fluid Dynamics Group
		The properties range from the very hot and dense plasmas of stars to the extremely diluted plasmas of the interstellar medium, which is only partially ionized.
		It is, however, a typical property of astrophysical plasmas, that the dynamics of magnetic fields is alternating between an ideal motion, where all forms of knottedness and linkage of the field are conserved (topology conservation), and a kind of disruption of the magnetic structure, the so called magnetic reconnection.
		Investigation of topological properties of magnetic and electromagnetic fields and their dynamics, especially with respect to critical phenomena like magnetic reconnection.
	www.tp4.ruhr-uni-bochum.de /vw/project.html (966 words)

	Topological property - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.
		That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property.
		A common problem in topology is to decide whether two topological spaces are homeomorphic or not.
	en.wikipedia.org /wiki/Topological_property (1301 words)

	Automated searching of crests and talwegs in DTTM/DTM GIS basis for environmental surveying
		Secondly, a graph is extracted of a given graph, using digital properties and mainly topological properties of this graph.
		This generation uses the properties of the Digital and Topological Terrain Model object to locate it in its physical context (spatial position) and structural context (its position in comparison to other objects).
		This study is a recognition of properties and shapes on a map represented by its graph.
	libraries.maine.edu /Spatial/gisweb/spatdb/egis/eg94042.html (3282 words)

	James Tauber : Poincare Project: Topological Properties Revisited
		Recall that a topological property is one based only on the open sets of a topology and not any other structure.
		For this reason a topological property is preserved under a homeomorphism.
		If one topological space has a topological property and another doesn't have that property then the two spaces can't be homeomorphic.
	jtauber.com /blog/2005/04/27/poincare_project:_topological_properties_revisited (175 words)

	BioMed Central \| Full text \| Structural comparison of metabolic networks in selected single cell organisms
		If these metabolic network properties of the organisms studied can be extended to other species in their respective domains (which is likely), then the design principle(s) of Archaea are fundamentally different from those of Bacteria and Eukaryote.
		Specifically, the drastic differences of topological profiles between the metabolic networks of Archaeal species and non-Archaeal species may be partially explained by the fact that significantly less extensive metabolic pathway studies have been conducted in Archaeal species [32].
		Robustness of topological profiles against random perturbations can alleviate the impact to a certain extent but is unable to eradicate it [9].
	www.biomedcentral.com /1471-2105/6/8 (4956 words)

	BioMed Central \| Full text \| Characterizing disease states from topological properties of transcriptional regulatory ...
		Recent advances in network analysis have focused on topological changes and static and dynamical network properties in yeast and E coli [20,21].
		The differences between these networks are reflected in their topological properties, e.g., the centrality of each gene in the network, the activity status of each link in the network, the Hamming distance between networks, to name just a few.
		The classification schemes introduced here utilize the topological properties of the network and facilitate the identification of key transcription factors that may be involved in gene dysregulation.
	www.biomedcentral.com /1471-2105/7/236 (8185 words)

	Good Math, Bad Math : Big to Small, Small to Big: Topological Properties through Sheaves (part 1)
		So far, in our discussion of topology, we've tended to focus either very narrowly on local properties of T (as in manifolds, where locally, the space appears euclidean), or on global properties of T....
		To explore the category theory a little further: once you've seen that a sheaf on a topological space is just a functor (presheaf) which respects certain limits (gluing properties I'm assuming will be described in a later post) there's no reason the course category has to be the poset of subspaces of a topological space.
		Okay, a subspace of a topological space can be identified with the continuous map from the subspace into the whole space.
	scienceblogs.com /goodmath/2006/12/big_to_small_small_to_big_topo.php (2743 words)

	Some Questions and References on Relative Topological Properties, Part 1
		Arhangel'skii A.V., Relative topological properties and relative topological spaces.
		Arhangel'skii A.V. and J. Tartir, A characterization of compactness by a relative separation property.
		Dow A., and J. Vermeer, An example concerning the property of a space being Lindelöf in another.
	at.yorku.ca /i/a/a/i/04.htm (728 words)

	DIMACS Workshop on Biomolecular Networks: Topological Properties and Evolution
		Relationships among gene sequences and gene expression patterns were resolved into networks and their topological properties were compared.
		Furthermore, the biological pathway of the cell-cycle sequence-which is a particular trajectory in the state space-is a globally stable and attracting trajectory of the dynamics.
		These dynamic properties, arising from the underlying network connection, are also robust against small perturbations to the network and against parameter changes in the model.
	dimacs.rutgers.edu /Workshops/Biomolecular/abstracts.html (2969 words)

	Topological Dynamics (Site not responding. Last check: 2007-11-03)
		The topological dynamics research group investigates topological and geometric properties of dynamical systems.
		Every nonsingular C1 flow on a closed manifold of dimension greater than two has a global transverse disk.
		Periodic Prime Knots in Topologically Transitive Flows on 3-Manifolds by Basener and Mike Sullivan of The University of Southern Illinois (To appear in Topology Appl.
	www.rit.edu /~kam1257/mathwebsite/topologicaldyn.htm (130 words)

	Properties of topological spaces
		A subset A of a topological space X is called closed if X - A is open in X.
		So the set of all closed sets is closed [!] under finite unions and arbitrary intersections.
		These four properties are sometimes called the Kuratowski axioms after the Polish mathematician Kazimierz Kuratowski (1896 to 1980) who used them to define a structure equivalent to what we now call a topology.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Lectures/L12.html (243 words)

	A Comparative Study of Topological Properties of Hypercubes and Star Graphs
		Undertakes a comparative study of two important interconnection network topologies:the star graph and the hypercube, from the graph theory point of view.
		[21] Y. Saad and M. Schultz, "Topological properties of hypercubes,"IEEE Trans.
		Citation: K. Day, A. Tripathi, "A Comparative Study of Topological Properties of Hypercubes and Star Graphs," IEEE Transactions on Parallel and Distributed Systems, vol. 05, no. 1, pp.
	doi.ieeecomputersociety.org /10.1109/71.262586 (529 words)

	DIMACS Workshop on Biomolecular Networks: Topological Properties and Evolution
		These points are then joined by edges to form a set of non-overlapping, irregular, space-filling tetrahedra each having the property that the sphere on the surface of which all four vertices reside does not contain a vertex from any other tetrahedron ("the empty sphere property").
		Additionally, analysis of the graph properties of real and model structures shows that D determines how the characteristic path length of the contact graph (the average inter-residue graph distance between all pairs of residues) scales with the number of residues.
		Here we model a protein structure as a network where nodes represent residues and unweighted edges the presence of interactions between residues.
	dimacs.rutgers.edu /Workshops/Biomolecular/posters.html (2746 words)

	Topological properties of spaces ordered by preferences
		In this paper, we analyze the main topological properties of a relevant class of topologies associated with spaces ordered by preferences (asymmetric, negatively transitive binary relations).
		This class consists of certain continuous topologies which include the order topology.
		The concept of saturated identification is introduced in order to provide a natural proof of the fact that all these spaces possess topological properties analogous to those of linearly ordered topological spaces, inter alia monotone and hereditary normality, and complete regularity.
	www.hindawi.com /GetArticle.aspx?doi=10.1155/S0161171299220170 (112 words)

	TOPOLOGICAL METHODS IN GROUP THEORY
		First, it is for graduate students who have had an introductory course in algebraic topology and who need a bridge from common knowledge to the current research literature in geometric and homological group theory.
		Secondly, I am writing for group theorists who would like to know more about the topological side of their subject but who have been too long away from topology.
		Thirdly, I hope the book will be useful to manifold topologists, both high- and low-dimensional, as a reference source for basic material on proper homotopy and homology..."
	www.math.binghamton.edu /ross/contents.html (341 words)

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