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Topic: Alexandrov topology

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	Alexandrov topology at AllExperts
		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.
		Alexandrov spaces were also rediscovered around the same time in the context of topologies resulting from denotational semantics and domain theory in computer science.
		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.allexperts.com /e/a/al/alexandrov_topology.htm (1379 words)

	Topology glossary at AllExperts
		;Alexandrov topology: A space X has the Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed.
		Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
		Weak topology: 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.allexperts.com /e/t/to/topology_glossary.htm (4523 words)

	Pavel Sergeevich Alexandrov Summary
		There, he and Uryson began their study of the new field of topology, the branch of mathematics that deals with properties of figures related directly to their shape and invariant under continuous transformation (that is, without cutting or tearing).
		Alexandrov's studies took him to the University of Moscow, where he became known as the leader of the Soviet topologists.
		Alexandrov went to a Moscow State University where he was a student of Dmitri Egorov and Nikolai Luzin.
	www.bookrags.com /Pavel_Sergeevich_Alexandrov (1409 words)

	Topology
		Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.
		Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular computational complexity theory.
		In topology a boundary component of a compact surface is a connected component consisting of boundary points of a surface.
	www.shortopedia.com /T/O/Topology (1036 words)

	BioGeometry Recommended Literature
		It is often said that topology started as a focused discipline with the work of Henri Poincare right before and after the end of the nineteenth century.
		Combinatorial studies in topology lead naturally to algebraic structures as discrete representations of continuous spaces.
		The algebraisation of topology initiated by Emmy Noether has led to a field dominated by the algebraists.
	biogeometry.duke.edu /education/literature/topology.html (416 words)

	Aleksandrov biography
		This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander.
		Today the Department of General Topology and Geometry of Moscow State University is Russia's leading centre of research in set-theoretic topology.
		After Aleksandrov's death in November 1982, his colleagues from the Department of Higher Geometry and Topology, in which he had held the chair, sent a letter to Moscow University's rector A A Logunov proposing that one of Aleksandrov's former students should become Head of the Department, to preserve Aleksandrov's scientific school.
	www-history.mcs.st-andrews.ac.uk /Biographies/Aleksandrov.html (2252 words)

	Chu Spaces (Site not responding. Last check: 2007-10-20)
		However when adding topology to relational structures, the topology cannot be incorporated into the relational structure but must continue to use open sets.
		Topology is more expressive than order in that it permits distinctions to be drawn that order alone cannot.
		When the open sets are closed under arbitrary intersection, as with the Alexandrov topology of a poset, it ceases to be possible to link the fate of one point under a function to that of an infinite sequence of points.
	chu.stanford.edu (5453 words)

	Interactive Mathematics Miscellany and Puzzles
		This little book is intended for those who desire to obtain an exact idea of at least some of the most important of the fundamental concepts of topology but who are not in a position to undertake a systematic study of this many-sided and sometimes not easily approached science.
		Of course, one cannot learn topology from these few pages; if however, one gets from them some idea of the nature of topology-at least in one of its most important and applicable parts, and also acquires the desire for further individual study-then my goal will have been reached.
		From this point of view let me direct those of you who already have the desire to study topology to the book written by Herr Hopf and myself which will soon be printed by the same publisher [see footnote 4-A.E.F.I.].
	www.cut-the-knot.org /books/alexandrov/forward.shtml (305 words)

	Professor A (Site not responding. Last check: 2007-10-20)
		Makarov proved, by means of geometry, that the number of non-isomorphic Fedorov groups in the Lobachevsky space is infinite (the similar fact on the plane was known already in XIX century).
		In 1963 three students began post-graduate studies in the field of geometry and topology under the supervision of A. Zamorzaev.
		Among many problems of the development of geometry discussed at the conference the discussion on the teaching of geometry was the most heated and lively; the tone of discussion was set by Academician A. Alexandrov, who that time was actively engaged in preparing new school textbooks in geometry.
	www.mi.sanu.ac.yu /vismath/zam (4327 words)

	Topology
		An undergraduate introduction to the fundamentals of topology — engagingly written, filled with helpful insights, complete with many stimulating and imaginative exercises to help students develop a solid grasp of the subject.
		Comprehensive coverage of elementary general topology as well as algebraic topology, specifically 2-manifolds, covering spaces and fundamental groups.
		Over 140 examples, preceded by a succinct exposition of general topology and basic terminology.
	www.doverdirect.com /0486656764.html (283 words)

	Re: Topology, Analysis, and Relativity (Site not responding. Last check: 2007-10-20)
		The Alexandrov topology coincides with the manifold one if and only if the space-time is strongly causal, i.e.
		Incidentally, it's interesting to note that an equivalent condition is that the Alexandrov topology be Hausdorff.
		The canonical reference for much of this stuff is Penrose's 'Techniques of differential topology in relativity' which was published by SIAM in about 1972.
	www.lns.cornell.edu /spr/2001-04/msg0032661.html (231 words)

	Amazon.ca: The Large Scale Structure of Space-Time: Books: S. W. Hawking,G. F. R. Ellis (Site not responding. Last check: 2007-10-20)
		The more interesting topics discussed in this chapter include the causality conditions (there are no closed non-spacelike curves), and the Alexandrov topology and its connection with the strong causality condition (every neighborhood of a point contains a neighborhood of the point no non-separable curve of which intersects it more than once).
		When strong causality does hold, the Alexandrov topology is equivalent to the usual manifold topology, and thus the topology of spacetime can be determined by the observation of causal relationships.
		It covers in brilliant form the gravitational collapse of a star, the theory of fl holes, the space-time singularities, the causal structure of space-time, and in its end the initial singularity of the universe, popularly known as the Big Bang.
	www.amazon.ca /Large-Scale-Structure-Space-Time/dp/0521099064 (1951 words)

	PlanetMath: Alexandrov one-point compactification
		The Alexandrov one-point compactification of a non-compact topological space
		one-point compactification, Alexandroff one-point compactification, Aleksandrov one-point compactification, Alexandrov compactification, Aleksandrov compactification, Alexandroff compactification
		This is version 6 of Alexandrov one-point compactification, born on 2003-07-27, modified 2005-02-06.
	planetmath.org /encyclopedia/AlexandrovOnePointCompactification.html (73 words)

	Gokova '01
		Gromov's compactness theorem is one that states that the collection of manifolds with Ricci curvature bounded from below, and diameter bounded from above is precompact in a suitable topology.
		The limiting objects are examples of Alexandrov spaces.
		In particular, it appears that it may be possible to use gauge theory to say something aboutcertain Alexandrov spaces.
	arf.math.metu.edu.tr /~gokova/2001 (268 words)

	Warwick: DCS: Reports and Theses
		The T0 world of Scott's topological models used in the denotational semantics of programming languages may at first sight appear to have nothing whatever in common with the Hausdorff world of metric space theory.
		Using our "partial metric" we introduce a new approach by constructing each semantic domain as an Alexandrov topology "sandwiched" between two metric topologies.
		S.G. Matthews, "Partial Metric Topology", Papers on General Topology and Applications, Annals of the New York Academy of Sciences 728, ed.
	www.dcs.warwick.ac.uk /reports/222.html (238 words)

	lectures.html
		Some interesting classes of posets, such as face lattices of convex polytopes, lattices of subgroups, and posets of words, are discussed.
		We study the cohomology of the Alexandrov topology underlying a partially ordered set P with coefficients in a sheaf S of rings.
		Algebraic topology is an algebraic formalism which allows one to make precise various kinds of qualitative information (called briefly "connectivity information") about geometric objects, even in high dimensions.
	math.sfsu.edu /gubeladze/cbms/lectures.html (1863 words)

	Amazon.ca: Convex Polyhedra: Books: A.D. Alexandrov,N.S. Dairbekov,S.S. Kutateladze,A.B. Sossinsky (Site not responding. Last check: 2007-10-20)
		by A.D. Alexandrov (Author), N.S. Dairbekov (Translator), S.S. Kutateladze (Translator), A.B. Sossinsky (Translator) "A polyhedron means a body bounded by finitely many polygons as well as a surface composed of finitely many polygons..." (more)
		A.D. Alexandrov was awarded the Stalin State Prize in 1942, the Lobachevsky prize in 1952, and the Euler Golden Medal in 1992.
		A polyhedron means a body bounded by finitely many polygons as well as a surface composed of finitely many polygons. Read the first page
	www.amazon.ca /Convex-Polyhedra-D-Alexandrov/dp/3540231587 (421 words)

	Wikinfo \| Partial order (Site not responding. Last check: 2007-10-20)
		That the class of Scott-Ershov domains is cartesian closed category enables the solution of so-called domain equations, e.g., D = [D -> D], where the right-hand side denotes the space of all continuous functions on D.
		Partially ordered sets can be given a topology, for example, the Alexandrov topology, consisting of all upwards closed subsets.
		For example, the natural topology on Scott-Ershov domains is the Scott topology.
	www.wikinfo.org /wiki.php?title=Partial_order (878 words)

	Counterexamples in Topology
		Over 25 Venn diagrams and charts summarize properties of the examples, while discussions of general methods of construction and change give readers insight into constructing counterexamples.
		Clear, introduction: graph imbedding, role of voltage graphs, Ringel-Youngs theorem, and genus of a group, including imbeddings of Cayley graphs.
		Excellent study of sets in topological spaces and topological vector spaces includes systematic development of the properties of multi-valued functions.
	store.doverpublications.com /048668735x.html (228 words)

	A Combinatorial Introduction to Topology
		By Subject > Science and Mathematics > Mathematics > Topology
		Excellent text for upper-level undergraduate and graduate students shows how geometric and algebraic ideas met and grew together into an important branch of mathematics.
		Suitable for courses in combinatorial computing and concrete computational complexity.
	store.doverpublications.com /0486679667.html (296 words)

	Modern mathematical proofs changing due to collaborations, computers
		The Poincare Conjecture, named after French mathematician Henri Poincare (1854-1912), states that a three-dimensional manifold with the homotopy of the sphere is the sphere.
		Krantz said that the task of validating the proof is so daunting that no single mathematician would be able to verify it because it demands the knowledge of difficult low-dimensional topology, Alexandrov theory — not well understood in the West — differential geometry and partial differential equations.
		But Perelman's indifference to publishing the proof and his method of showing his work on arXiv "have put a choke hold on the subject of low-dimensional topology," he said.
	www.eurekalert.org /pub_releases/2006-02/wuis-mmp013106.php (934 words)

	Topics: Topology, Topological Space
		Set of topologies on a set X: Given a set, the set of topologies on it is partially ordered by fineness; In fact, it is a lattice under inclusion, with meet
		On a metric/normed space: A topology is induced in any metric or normed space.
		On a Lorentzian manifold: Use the Alexandrov topology, or for compact cases Johan's definition.
	www.phy.olemiss.edu /~luca/Topics/t/top.html (583 words)

	M567: Boolean Algebra
		Determine the topological space Spec L for the distributive lattice L.
		A bounded lattice L is atomic if for each non-zero element x of L, there is an atom a of L such that a < x.
		is the Alexandrov topology on the specialization order of the space X ;
	orion.math.iastate.edu /jdhsmith/class/M567S05.htm (880 words)

	MASS - Research
		DIFFERENTIAL GEOMETRY AND TOPOLOGY OF CURVES AND SURFACES
		Alexandrov's Conjecture on Degenerate Surfaces and Surfaces of Revolution - James Krysiak
		The topology of orbits under a real flow -- Vivek Srikrishnan
	www.math.psu.edu /mass/research/index.html (1280 words)

	Topics: Spacetime and Topology
		Base: In a full chronological space, one is given by the Alexandrov neighborhoods {[x, y]} [@ Lerner in(72)].
		Special cases: Coincides with the manifold topology iff (M, g) is strongly causal (in which case it is Hausdorff), but in general it is coarser; In the discrete case it is trivial, and gives in general the discrete topology.
		Restrictions: There are none on the spatial topology for an asymptotically flat vacuum spacetime, although in most cases singularities will develop.
	www.phy.olemiss.edu /~luca/Topics/t/top_st.html (385 words)

	Amazon.fr : A.D. Alexandrov Selected Works: Intrinsic Geometry of Convex Surfaces: Livres en anglais: S. S. ... (Site not responding. Last check: 2007-10-20)
		Amazon.fr : A.D. Alexandrov Selected Works: Intrinsic Geometry of Convex Surfaces: Livres en anglais: S. Kutateladze,A. Aleksandrov,Kutateladze Kutateladze
		Aleksandrov, Kutateladze Kutateladze "Study of continuous length-preserving deformations of a surface is sensible not only for regular surfaces: it suffices that on a surface there be enough curves..." (plus)
		A.D. Alexandrov's contribution to the field of intrinsic geometry was original and very influential.
	www.amazon.fr /D-Alexandrov-Selected-Works-Intrinsic/dp/0415298024 (416 words)

	Amazon.com: Combinatorial Topology: Books: P. S. Alexandrov (Site not responding. Last check: 2007-10-20)
		Discover new releases in your favorite categories, popular pre-orders and bestsellers, exclusive author interviews and podcasts, special sales, and more.
		Fundamental topological facts, together with detailed explanations of the necessary technical apparatus, constitute this clearly written, well-organized three-part text.
		Part 1 deals with certain classic problems without using the formal techniques of homology theory; parts 2 and 3 focus on the central concept of combinatorial topology, the Betti groups.
	www.amazon.com /Combinatorial-Topology-P-S-Alexandrov/dp/0486401790 (654 words)

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