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Topic: Stronger topology

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Topology glossary

product topology

closure topology

completeness topology

Zariski topology

base topology

continuity topology

differential topology

weak topology

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neighborhood topology

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	Operator topology - Wikipedia, the free encyclopedia
		topology is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous.
		The weak topology is useful for compactness arguments as the unit ball is compact.
		The weak and strong topologies are widely used as cheap approximations to the ultraweak and ultrastrong topologies, and the remaining topologies are of little practical importance.
	en.wikipedia.org /wiki/Topologies_on_the_set_of_operators_on_a_Hilbert_space (983 words)

	Topology glossary - Wikipedia, the free encyclopedia
		If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced by T consists of all intersections of open sets in T with A.
		Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
		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.wikipedia.org /wiki/Topology_glossary (4669 words)

	54: General topology
		Topology is the study of sets on which one has a notion of "closeness" -- enough to decide which functions defined on it are continuous.
		Thus a general theme in topology is to test the extent to which the axioms force the kind of structure one expects to use and then, as appropriate, introduce other axioms so as to better match the intended application.
		Since the axioms of topology are stated in terms of subsets of X, it should be no surprise that one branch of topology is closely related to set theory, particularly "descriptive set theory".
	www.math.niu.edu /~rusin/known-math/index/54-XX.html (2431 words)

	Weak operator topology: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-18)
		Topology (greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces....
		In functional analysis, the strong operator topology, often abbreviated sot, is the weakest topology on the set of bounded operators on a hilbert space...
		In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest...
	www.absoluteastronomy.com /encyclopedia/w/we/weak_operator_topology.htm (779 words)

	Reference.com/Encyclopedia/Topologies on the set of operators on a Hilbert space
		In mathematics, the requirements of functional analysis mean there are several standard topologies which are given to the set of bounded linear operators on a Hilbert space.
		The norm topology is stronger than the strong operator topology which is stronger than the weak operator topology.
		The norm topology is stronger than the weak-star topology which is stronger than the weak operator topology.
	www.reference.com /browse/wiki/Topologies_on_the_set_of_operators_on_a_Hilbert_space (144 words)

	Topology glossary - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-18)
		Although there is no clear distinction between different areas of topology, this glossary focuses primarily on general topology and on definitions that are fundamental to a broad range of areas.
		A collection of open sets is a subbase (or subbasis) for a topology if every open set in the topology is a union of finite intersections of sets in the subbase.
		If B is any collection of subsets of a set X, the topology on X generated by B is the smallest topology containing B; this topology consists of all unions of finite intersections of elements of B.
	encyclopedia.learnthis.info /t/to/topology_glossary.html (3671 words)

	Mathematical Education: Stat'i S.P. Novikova
		I would like to note that, in the topology of the time, refined algebraic manipulations got tightly mixed up with topology of function spaces, cooperating with the geometry of manifolds, on the basis of fundamental ideas of transversality, cobordisms, and calculus of variations.
		Part II &mdash the differential approach to topology &mdash should be a course of topology, both understandable and useful, both for physicists and mathematicians working in different areas of analysis.
		A unification of topology with a world of other areas of science that are less abstract was the main goal of that part.
	www.mccme.ru /edu/index.php?ikey=n-buno (8657 words)

	Topology (Site not responding. Last check: 2007-10-18)
		A topology is a set of points, and a collection of subsets that are designated as "open".
		To be a proper topology, the empty set and the entire set must be open, and open sets remain open under finite intersection and arbitrary union.
		The maximum topology is stronger than all other topologies.
	www.mathreference.com /top,intro.html (403 words)

	Topology
		The family t is called a topology (for X) when it satisfies these axioms and its elements are called _open sets_ (open wrt the topology).
		The reader should now check that continuity in the sense of calculus of a function from R to R is equivalent to continuity as a map of topological spaces, with respect to the topology m.
		Call a topology t _stronger_ than the topology t' (both for the same set X) if t is contained in t'.
	www.georgetown.edu /faculty/kainen/topology.html (1132 words)

	55: Algebraic topology
		Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants reflect some of the topological structure of the spaces.
		General topology focuses on the underlying spaces and is often concerned with fairly analytical issues (e.g.
		The tools of algebraic topology, when developed in isolation or for applications to other fields such as ring theory, give rise to homological algebra and category theory; this is the proper framework for comparing different algebraic tools.
	www.math.niu.edu /~rusin/known-math/index/55-XX.html (2581 words)

	PlanetMath: quotient space
		is the topology whose open sets are the subsets
		that satisfies this stronger property is called a quotient map, and given such a quotient map, the space
		The topology on the quotient space is then chosen to be the strongest topology such that the projection map
	planetmath.org /encyclopedia/QuotientSpace.html (155 words)

	[No title]
		My theme is the transition from classical algebraic topology to stable algebr* aic topology*, with emphasis on the emergence of cobordism, K-theory, generalized ho- mology and cohomology, the stable homotopy category, and modern calculational techniques.
		In the first flowering of stable algebraic topology, with t* he introduction of cobordism and K-theory, the solidly established theory of fiber bundles was absolutely central to the translation of problems in geometric topo logy to problems in stable algebraic topology*.
		Stable and unstable homotopy groups Another important precursor of stable algebraic topology was a substantial in- crease in the understanding of the relationship between stable and unstable hom* *o- topy groups and of certain fundamental exact sequences relating homotopy groups in different dimensions.
	hopf.math.purdue.edu /May/history.txt (14491 words)

	METAPHOR, METONYMY, AND BINDING
		This is a metaphorical blend: input one has the stronger and weaker objects; input two has the contest between companies.
		The weaker and stronger objects in input one are concrete but not specific, and so cannot in themselves provide the corresponding specific elements in the blend.
		The printing press and car have topology in the blend (the press crushes and the car is crushed) that their counterparts in Input 2 do not have (the press is an instrument of making newspapers and the car is a salient product of the automobile company).
	markturner.org /metmet.html (4532 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)

	Amazon.com: Topology of Surfaces, Knots, and Manifolds: Books: Stephan C. Carlson (Site not responding. Last check: 2007-10-18)
		Topology of Surfaces, Knots, and Manifolds offers an intuition-based and example-driven approach to the basic ideas and problems involving manifolds, particularly one- and two-dimensional manifolds.
		A blend of examples and exercises leads the reader to anticipate general definitions and theorems concerning curves, surfaces, knots, and links--the objects of interest in the appealing set of mathematical ideas known as "rubber sheet geometry." The result is a book that provides solid coverage of the mathematics underlying these topics.
		This book presents the topology of surfaces, manifolds and knots in a manner that is reachable for undergraduate students with only a knowledge of calculus.
	www.amazon.com /exec/obidos/tg/detail/-/0471355445?v=glance (1156 words)

	APPENDIX A
		A norm determines the norm topology by the open ball neighborhoods of x with radius a.
		The open sets of the norm (uniform) topology of Alg are once again, generated by the open balls of the induced norm.
		The norm topology is stronger (= finer = "has more open sets" = "fewer convergent sequences") than the strong operator topology.
	graham.main.nc.us /~bhammel/SPDER/apdxA.html (1576 words)

	[No title]
		Since it turns out that the stronger equality can in fact be obtained with essentially the same proof as in [33], we now present the result directly in that form.
		This is the topology for which a basis of open sets consists of all sets of the form Q1i=1 Ui, where each Ui is an open subset of IRm and only finitely many of them are proper subsets of IRm.
		Moreover, these conclusions can be established as consequences of a more general result about convergent generating series, which insures that there exists a generic subset W of IRm;1 with the property that these jets suffice for distinguishing all possible convergent generating series; details are given in the upcoming paper [32].
	www.math.rutgers.edu /~sontag/orders-yw-siam2.html (11061 words)

	Compact and Hausdorff
		Making a topology weaker keeps a space compact, for any open cover is an open cover in the original compact space.
		We already showed that a map from a compact space onto a hausdorff space is a homeomorphism, and that isn't the case here; hence the domain of f is not compact.
		Similarly, when g maps s onto itself with a weaker topology, g is not bicontinuous, and the range is not hausdorff.
	www.mathreference.com /top-cs,haus.html (693 words)

	THE DENTABILITY IN THE SPACES WITH TWO TOPOLOGIES (Site not responding. Last check: 2007-10-18)
		Gjinushi has given a local version of the first lemma in a normed vector space [3] and E. Saab [7] has proved that this lemma is true in a barreled locally convex Hausdorff space.
		Here it will be proved that the first lemma of Phelps is true in every topological vector space supplied with two comparative topologies; the other lemma as well will be proved in locally convex (topological) spaces with two comparative topologies.
		Two theorems are obtained for r-dentability in a local vector space with two topologies in which the stronger topology is that of the type B.M. Then these two theorems imply six known theorems for dentability and w
	math.la.asu.edu /~rmmc/rmj/VOL27-4/FUND (190 words)

	22nd Annual Workshop in Geometric Topology
		We will discuss a class of aspherical manifolds where this conjecture is true; namely those that support Riemannian metrics of non-positive sectional curvature (and dim >4).
		We show how a somewhat stronger version of this result implies that compact complete affine flat manifolds are determined up to homeomorphism by their fundamental group.
		Identify yourself as being with the "Geometric Topology Workshop Block - June 8-11".
	www.coloradocollege.edu /dept/ma/Topology1 (1059 words)

	Probability Abstract Service
		We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least 3, and the net transition on each edge is bounded by e.
		Our results are the first rigorous results to establish the role of phase transitions for markov random fields on trees as studied in probability, statistical physics and information theory to the study of phylogenies in mathematical biology.
		We extend his main-result, the Universal Limit Theorem, to a stronger Hoelder topology.
	www.math.u-psud.fr /~paserv/Letters/letter_75.shtml (8354 words)

	AMCA: Duality as a Unifying Framework by Ingrid Rewitzky, Hilary Priestley, Mai Gehrke, Achim Jung, Marcello Bonsangue
		Here several topologies, including the interval topology as well as a stronger topology, special to canonical extensions, play a fundamental role: maps are extended using analogues of the well-known liminf and limsup constructions from real analysis.
		This research was motivated by Domain Theory, in the sense of Scott's approach to Denotational Semantics of programming languages, and indeed many classes of domains are subsumed by the general construction.
		However, the primary structure of domains is order, not topology, and the challenge to capture this setting accurately in Stone duality is still open.
	at.yorku.ca /c/a/o/p/07.htm (954 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		A topology stronger than the usual topology is constructed on the set of distributions over a finite or countably infinite alphabet.
		For this topology, convergence P_n to P for an ordinary sequence is the same as convergence of the divergence D(P_nP) to 0.
		It is hoped that clickable material will be available here before the workshop.
	www.math.ku.dk /IT-Banach2001/harremoesoct.html (179 words)

	[No title]
		We identify a topology (induced by what we call d^*) with respect to which the associated undominated Bayesian Nash equilibrium (UBNE) correspondence is upper hemi-continuous at any complete information prior.
		As a corollary, we show that almost any social choice function is robustly UNE implementable relative to d^*.
		Moreover, we show that only monotonic social choice functions can be robustly UNE implementable relative to what we call d^{} which induces a slightly stronger topology** than that induced by d^*.
	eswc2005.econ.ucl.ac.uk /ESWC/2005/Prog/viewpaper.asp?pid=212 (118 words)

	Atlas: Operator representations for spaces of vector valued holomorphic functions by Klaus D. Bierstedt (Site not responding. Last check: 2007-10-18)
		For weighted spaces HV(G) of holomorphic functions with a topology stronger than uniform con-
		des topologies'' and to the vector valued projective description problem.
		A positive result is derived, and some examples are given.
	atlas-conferences.com /cgi-bin/abstract/cacl-52 (165 words)

	Untitled Document (Site not responding. Last check: 2007-10-18)
		This structure is then exploited to prove probabilistic limit theorems.
		We establish a large deviations principle (LDP) for the empirical measures, in a topology stronger than the usual tau topology, and with rate-function given explicitly in terms of relative entropy.
		Moreover, this LDP can be refined to a precise expansion similar to the Bahadur-Ranga Rao expansion for independent random variables.
	math.stanford.edu /~applmath/fall03/yiannis.htm (195 words)

	Unix:
		This new topology, while slightly more constrained, still treated files and directories roughly equally, true to the ethos of the project.
		Having moved past the question of whether one user was able to use the system effectively, it became time to test the system as it was intended be used, with many users connected at once making read and write calls to the disk simultaneously.
		One aspect of this approach that made their work more difficult was that they had to be very careful that all the parts would work together; when many small pieces are put together to form a finished project, the interaction between the pieces can often lead to mistakes.
	www.princeton.edu /~hos/frs122/unixhist/finalhis.htm (19978 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		A topology stronger than the usual product topology is necessary for uniform control over the sticks to the left of the origin and because $Y$ is not even a closed subset of $[0,\infty)^\mmZ$.
		Recall that the Radon measures on $\mmR$ are those nonnegative Borel measures under which bounded sets have finite measure, and that their vague topology is defined by declaring that $\nu_n\to\nu$ if $\nu_n(\phi)\to\nu(\phi)$ for all functions $\phi \in C_0(\mmR)$ (compactly supported, continuous).
		0$, $\a^N_{Nt} \to u(x,t)dx$ in probability as $N\to\infty$, in the vague topology of Radon measures on $\mmR$.
	www.ma.utexas.edu /mp_arc/html/papers/95-403 (6289 words)

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