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Topic: Weaker topology

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algebraic topology

Topology glossary

product topology

closure topology

completeness topology

Zariski topology

base topology

continuity topology

differential topology

weak topology

general topology

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	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)

	Topological space - Wikipedia, the free encyclopedia
		Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
		The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety.
		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.
	en.wikipedia.org /wiki/Topological_space (1750 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)

	Mathematics - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-03)
		Topology connects the study of space and the study of change by focusing on the concept of continuity.
		Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for doing just that.
		Topology -- Geometry -- Trigonometry -- Algebraic geometry -- Differential geometry -- Differential topology -- Algebraic topology -- Linear algebra -- Fractal geometry
	encyclopedia.learnthis.info /m/ma/mathematics.html (2172 words)

	Dsl High Speed Internet Service (Site not responding. Last check: 2007-11-03)
		For suppose not; then the quotient topology on X is the (set-theoretic) disjoint union topology, topological sum of the subject area.
		In other words, the fixed pointss of this is the union of {Xi}, then the quotient topology on A induced by f is a neighbourhood of a set of the polynomials x1-c1,..., xn-cn.
		A space is a subcover (or subcovering) of a space X is a set is the coarsest topology on Y induced by d.
	t1.2vv1.com /dslhighspeedinternetservice.html (1495 words)

	Novell's Networking Primer
		The distributed star topology, illustrated in Figure 13, is a more complex form of the physical star topology, with multiple central connection points connected to form a string of stars.
		However, the star topology and its derivatives are also susceptible to bottlenecking and single-point failure; the latter is often remedied by providing a redundant backup of the hub node.
		Because the logical topology is associated with the path and direction of data, it is closely linked with the MAC methods in the media access layer of the OSI model.
	www.novell.com /info/primer/prim08.html (3973 words)

	Mathematics - Wikipedia, the free encyclopedia
		The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology.
		Topology in all its many ramifications may have been the greatest growth area in 20
		Understanding and describing change is a common theme in the natural sciences, and calculus was developed as the most useful tool.
	en.wikipedia.org /wiki/Mathematics (3554 words)

	Traditional Topology Tools (Site not responding. Last check: 2007-11-03)
		The size of topology data in the cache file will be approximately the same as the size of the producing drawing, excluding the size of the drawing's table.
		Stored topology offers performance advantages when computing topology overlays for large data sets on weaker computer systems, but at the cost of significant limitations compared to topology on the fly.
		Stored topology is an effective approach for single-threaded operation but it is very poorly suited to multi-threaded, parallel or distributed operations using multiple processors.
	www.manifold.net /doc/7x/traditional_topology_tools.htm (1317 words)

	Customer Dsl Service (Site not responding. Last check: 2007-11-03)
		Compact-open topology The compact-open topology on M; this is the countable union of two sets is a base for its topology.
		A space X is the finest topology for which all the injection maps are continuous.
		Two continuous maps between two spaces X and Y is a base (or basis) for a topology on the set S. A point x is a subset of some terms used in MRIss to distinguish between tissue types.
	t1.2vv1.com /customerdslservice.html (1976 words)

	[No title]
		The relative weak topology on $\al G.$ seems to be easier for construction of models where the action of $K$ on $\al F.$ is continuous.
		Moreover this topology is not obviously natural from the viewpoint of gauge groups acting by automorphisms of operator algebras.
		The relative weak topology on the group $C(X,U(n))$ is more interesting, because it is actually finer than both the Bohr topology and the topology of convergence in measure.
	www.ma.utexas.edu /mp_arc/html/papers/04-14 (3467 words)

	PMTH335 - Topology
		OURSE IN Topology is one of the newer branches of mathematics, its origins going back to about the turn of the century.
		It is then easy to see that two topologies on the same underlying set are ``essentially the same'' if and only if the same functions are continuous with respect to one of these topologies as are continuous with respect to the other.
		Many problems in topology are concerned with whether properties of given topological spaces or functions defined in terms of them are preserved by such constructions.
	mcs.une.edu.au /~pmth335/index2.html (1758 words)

	Topology Reviews (Site not responding. Last check: 2007-11-03)
		The reader can see clearly that the weaker the topology on a space the harder it is for mappings to be continuous on the space.
		Recognizing that the only functions able to be continuous on a space with the indiscrete topology are the constants, and that a space with the discrete topology has continuous functions in abundance, the author asks the reader to consider topologies that fall between these extremes, and this motivates the separation properties of topological spaces.
		Ones first exposure to algebraic topology should be a concrete and pictorial approach to gain a visual and combinatorial intuition for algebraic topology.
	www.booksunderreview.com /Science/Math/Topology (7449 words)

	Algebraic Topology: Topology
		The topology on A defined by F is the weakest topology (i.e., the smallest collection OA) for which all these functions become continuous.
		The topology on B defined by F is the strongest topology (i.e., the largest collection OB) for which all these functions become continuous.
		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.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	Guide to the Mathematics Subject Classification Scheme
		The second broad part of the mathematics literature includes those areas which could be considered either independent disciplines or central parts of mathematics, as well as those areas which clearly use mathematics but are interested in non-mathematical ideas too.
		Here the general theory is much weaker, but special cases of such rings are of key importance: Lie algebras in particular, as well as Jordan algebras and other types.
		55: Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants illustrate some of the rigidity of the spaces.
	www.math.niu.edu /~rusin/known-math/index/beginners.html (5525 words)

	categories: Re: Abelian Topological Groups (Site not responding. Last check: 2007-11-03)
		Moreover, although a weaker topology (or an abelian group with a weaker topology, which is what I assume is meant) is certainly a subobject, it is not regular, which every subobject in an abelian category must be.
		The idea is that the > quotients of such a group, in the abelian category, would be completions > of the group with respect to topologies coarser than the given one.
		Of course, having a topology as an > object in the abelian category means we have to have objects in the category > other than abelian groups.
	north.ecc.edu /alsani/ct01(5-8)/msg00001.html (352 words)

	Amazon.com: Topology (2nd Edition): Books: James Munkres (Site not responding. Last check: 2007-11-03)
		I must say I was quite confused when I began the actual topology portion of the book (chapter 2), but this was due to the difficulty of topology, in general, and not the book itself.
		Unfortunately, this lack of a completely cohesive approach is unavoidable, since a course in point-set topology ought to provide a stepping stone one can use for further study in topology and not a mountain one can climb and conquer and thus know the subject completely.
		If you are searching for an introduction to point-set topology that will give you a solid grounding in the basics of point-set topology, but at the same time will give it to you in an easily approached manner, than this book is for you.
	www.amazon.com /Topology-2nd-James-Munkres/dp/0131816292 (1691 words)

	III. FCCR LIMITS
		A sufficient condition for a bounded operator not of finite rank to be of trace class, is that the spectral values have only one accumulation point, that it be zero, and that convergence to zero is sufficiently rapid that the sum analogous to (3.10) be bounded.
		A simple convergence in the norm topology is impossible since B, B!, and N are all unbounded operators.
		A picture of the convergence is that the effect of this element is damped in the limit to combat the nascent singularity, and that the contributing edge effects eventually "fall off the edge" in a limit weaker than the norm topology.
	graham.main.nc.us /~bhammel/FCCR/III.html (2357 words)

	Zariski Topology
		The zariski topology is consistent with the metric topology, though it is weaker.
		Let k be a field with a valuation metric, such as the p-adic numbers, or the fraction field of any pid, or the completion thereof.
		Since addition and multiplication are continuous, polynomials are continuous, and within this context the zariski topology is consistent with the metric topology, but weaker, as shown by the powers of p in k
	www.mathreference.com /ag,zar.html (544 words)

	pub2 (Site not responding. Last check: 2007-11-03)
		If X is sigma-compact Polish, then Ck(X) has a sigma-closure-preserving base, Topology Appl., 151(2005), 99-106 (with K. Tamano).
		Metrizable spaces and generalizations, in Recent Progress in General Topology II, M. Husek and J. van Mill, eds., Elsevier, Amsterdam, 2002, pp.
		Weaker connected and weaker nowhere locally compact topologies for metrizable and similar spaces Topology Appl.
	www.auburn.edu /~gruengf/recentpaps.html (203 words)

	Second Galway Topology Colloquium Proceedings: Manifolds At and Beyond the Limit of Metrisability by David Gauld (Site not responding. Last check: 2007-11-03)
		Some of these conditions are strictly weaker, some strictly stronger and others unrelated to metrisability in a general topological space.
		The main tool from outside topology which is used to study large spaces in general and non-metrisable manifolds in particular is Set Theory.
		Recently some techniques of Algebraic Topology have been combined with ideas from Set Theory to determine the torsion of the group of homeomorphisms of powers of the long line.
	webdoc.sub.gwdg.de /ebook/e/2000/at.yorku.ca/98/24.htm (222 words)

	Topology!!!!
		-open interval topology is separable - the rational numbers are dense in this topology.
		is uncountable, and this topology is not second countable.
		be equipped with the topology of pointwise convergence.
	www.math.unl.edu /~s-bbockel1/922-notes/node7.html (388 words)

	Introduction
		Such a visualization is in effect a topology preserving map of the underlying data collection.
		The proximity of a pair of objects from a data collection can be expressed either as their similarity, mutual agreement, or dissimilarity - their distance in the abstract domain of the collection.
		An ordinal scale is weaker than quantitative, in that it also induces an ordering of objects, but does not make any statement about the magnitude of the differences.
	www.pavis.org /essay/introduction.html (2867 words)

	Definition and examples of topologies
		It is the topology associated with the discrete metric.
		The discrete topology is the strongest topology on a set, while the trivial topology is the weakest.
		is a topology in which, for example, the interval (0, 1) is not an open set.
	www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L11.html (258 words)

	M.G. Tka\v{c}enko, V.V. Tkachuk, V.V. Uspenskii, R.G. Wilson (Site not responding. Last check: 2007-11-03)
		Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology.
		It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors.
		\par The requirement that a space have a weaker Tychonoff connected topology is rather strong, but we show that it is difficult to construct spaces which would contain no infinite subspaces with a weaker connected $T_{3{1\over 2}}$-topology.
	www.univie.ac.at /EMIS/journals/CMUC/cmuc9604/abs/tkauswil.htm (140 words)

	Abstract Stone Duality (Site not responding. Last check: 2007-11-03)
		It illuminates the duality between open and closed concepts that is obscured by asymmetry in classical and intuitionistic topology.
		The constructive and conceptual traditions have long been in conflict, but ASD is well on the way to reconciling them.
		Besides further outreach into analysis, future work will consolidate ASD into a "type theory" usable by ordinary mathematicians, that automatically gives them the topology on and programs for their constructions.
	www.cs.man.ac.uk /~pt/ASD (541 words)

	Mereology
		And since (34) follows from (P.5), it might be concluded that EM should be rejected in favor of the weaker mereological theory MM.
		First, it could be observed that the ontological exhuberance associated with the relevant closure principles is not substantive -- that the increase of entities in the domain of quantification of a closure mereology involves no substantive additional commitments besides those already involved before the closure.
		Forrest, P., 1996a, ‘From Ontology to Topology in the Theory of Regions’, The Monist 79: 34-50.
	plato.stanford.edu /entries/mereology (9670 words)

	Volume 22, Number 3, 1996
		Then we highlight the connections between set convergence, with respect to the slice and Attouch-Wets topologies, and convergence, in the same sense, of the associated functions.
		Finally, by using known results on the behaviour of the subdifferential of a convex function under the former epigraphical perturbations, we are able to derive stability results for the set of supported points and of supporting and exposing functionals of a closed convex subset of a Banach space.
		Under some assumption of fast enough convergence on the sequence of ("almost" nonexpansive) perturbed iteration mappings, if the basic method is $\tau$-convergent for a suitable topology $\tau$ weaker than the norm topology, then the perturbed method is also $\tau$-convergent.
	www.math.bas.bg /~serdica/n3_96.html (1010 words)

	Time Travel [Internet Encyclopedia of Philosophy]
		In this article, we make a distinction between time travel stories that might be possible within the canon of known physical laws and those stories that contravene or go beyond known laws.
		According to the principle of equivalence, then, a clock at sea level on the Earth runs a little slower than a clock at the top of Mount Everest because the strength of the field is weaker the further you are from the center of mass.
		While most of spacetime seems to be flat or gently rolling contours, physicists are aware of spacetime regions with unusual and severe topologies such as rotating fl holes.
	www.iep.utm.edu /t/timetrav.htm (7856 words)

	Citebase - On Tychonoff-type hypertopologies (Site not responding. Last check: 2007-11-03)
		In 1975, M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it.
		The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family M of subsets of a set X. When M contains all singletons, a description of all Tychonoff-type topologies O on M is given.
		The problem of commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology} is investigated as well.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0204121 (528 words)

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