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Comparison of topologies - Wikipedia, the free encyclopedia All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology. The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology. en.wikipedia.org /wiki/Finer_topology   (496 words)

Topologies on the set of operators on a Hilbert space - Wikipedia, the free encyclopedia These topologies are all locally convex, which implies that they are defined by a family of seminorms. The ultraweak and ultrastrong topologies are better in some ways than the weak and strong topologies, but their definitions are more complicated, so they are usually not used unless their better properties are really needed. 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   (904 words)

Finer topology   (Site not responding. Last check: 2007-11-06) The finest topology on X is the discrete topology. The coarsest topology on X is the trivial topology. Any two topologies on X have a meet and join, in the sense of lattice theory; the meet is the intersection, but the join is not in general the union. www.sciencedaily.com /encyclopedia/finer_topology   (288 words)

Weak topology - Wikipedia, the free encyclopedia The term is most commonly used for the initial topology of a normed vector space with respect to its (continuous) dual. The weak topology on X is the weakest topology (the topology with the fewest open sets) such that all elements of X If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space. www.wikipedia.org /wiki/Weak_topology   (369 words)

Box topology - Wikipedia, the free encyclopedia In general, the box topology is finer than the product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial). This does not always hold in the box topology, because it is in general a much finer topology, so therefore mapping into the range space makes it much harder for functions to be continuous. This actually makes the box topology very useful for providing counterexamples — many qualities such compactness, connectedness, metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product with this topology. www.wikipedia.org /wiki/Box_topology   (551 words)

topology 1 In mathematics, topology is a branch concerned with the study of topological spaces. Topology is also concerned with the study of the so-called topological properties of figures, that is to say properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms). In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories. www.fact-library.com /topology_1.html   (1191 words)

Strong operator topology - Wikipedia, the free encyclopedia In functional analysis, the strong operator topology, often abbreviated SOT, is the weakest topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number The SOT is stronger than the weak operator topology and weaker than the norm topology. The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus. en.wikipedia.org /wiki/SOT   (330 words)

[No title]   (Site not responding. Last check: 2007-11-06) Roughly speaking, a topology is a way of specifying the concept of "nearness"; an open set is "near" each of its points. The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety. 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. www.online-encyclopedia.info /encyclopedia/t/to/topological_space.html   (1204 words)

Topological space - Wikipedia, the free encyclopedia Any infinite set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. 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. For any such structure which is not finite, we often have a natural topology which is compatible with the algebraic operations in the sense that the algebraic operations are still continuous. en.wikipedia.org /wiki/Topological_space   (1574 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)

[No title] The topology P consists of the elements in O and the complements of closed subsets as neighborhoods of that point. Hausdorff topology +------------------------------------------------------------ The Hausdorff topology is a metric on the set of closed bounded subsets of a complete metric space. induced topology +------------------------------------------------------------ The induced topology on a subset A of X, where (X,T) is a topological spoace is the the topological space (A, Y cap A _Y in T). www.math.harvard.edu /~knill/sofia/data/topology.txt   (1652 words)

Strong topology - Wikipedia, the free encyclopedia In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: Note that a topology τ is stronger than a topology σ (is a finer topology) if it contains more open sets. en.wikipedia.org /wiki/Strong_topology   (128 words)

Science Fair Projects - Second-countable space   (Site not responding. Last check: 2007-11-06) In topology, a second-countable space is a topological space satisfying the "second axiom of countability". In general, the finer the topology, the less likely it is to be second-countable. The topology of a second-countable space has cardinality less than or equal to c (the cardinality of the continuum). www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Second-countable   (542 words)

Topological space -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06) Note that the requirement that the union of any collection of sets is a member of the topology is more stringent than simply requiring that all pairwise unions must be members, as the former includes unions of infinite collections of sets. A topology is completely determined if for every net in X the set of its (Click link for more info and facts about accumulation point) accumulation points is specified. The (Click link for more info and facts about Zariski topology) Zariski topology is a purely algebraically defined topology on the (Click link for more info and facts about spectrum of a ring) spectrum of a ring or an (Click link for more info and facts about algebraic variety) algebraic variety. www.absoluteastronomy.com /encyclopedia/t/to/topological_space.htm   (3343 words)

Box topology -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06) In (The configuration of a communication network) topology, the (The set of elements common to two or more sets) cartesian product of ((mathematics) any set of points that satisfy a set of postulates of some kind) topological spaces is can be topologized in several ways. The canonical way of doing it is using the (Click link for more info and facts about product topology) product topology, because it fits rather nicely with the (Click link for more info and facts about categorical) categorical notion of a (An artifact that has been created by someone or some process) product. It is easily verified that B is actually a (The fundamental assumptions from which something is begun or developed or calculated or explained) basis for the topology. www.absoluteastronomy.com /encyclopedia/B/Bo/Box_topology.htm   (548 words)

PlanetMath: topological space It is the smallest or coarsest possible topology on it might be interesting to note that you can equivalently define a topology in terms of it's closed sets, by demorgan's set laws. i can't seem to find a definition for the topology induced by a metric space either, which is something you probably want to add. planetmath.org /encyclopedia/Topology.html   (168 words)

PlanetMath: Hausdorff space not completely Hausdorff In general, in the finer topology, all basic sets were both open and closed. We will use the closed-neighborhood sense for completely Hausdorff, which will also imply the topology is not completely Hausdorff in the functional sense. This proves the topology under consideration is not completely Hausdorff (under both usual meanings). planetmath.org /encyclopedia/HausdorffSpaceNotCompletelyHausdorff.html   (326 words)

Topological space   (Site not responding. Last check: 2007-11-06) A topology is completely determined if for every net in X the set of its accumulation pointss is specified. The set of real numbers R is a topological space: the open sets are generated by the base of open intervals. A space carries the trivial topology if all points are "lumped together" in the sense that there are only two open sets, the empty set and the whole space. www.sciencedaily.com /encyclopedia/topological_space   (2157 words)

Finer topology   (Site not responding. Last check: 2007-11-06) The finest topology on X is the discrete topology.The coarsest topology on X is the trivial topology. Anytwo topologies on X have a meet and join, in the sense of lattice theory ; the meet is the intersection, but the join is not in general the union. See topologies on the set of operators on a Hilbert space for someintricate relationships. www.therfcc.org /finer-topology-219338.html   (234 words)

Books of Topology - Amzp.com   (Site not responding. Last check: 2007-11-06) In chapter 3, the author considers first the topology on the space of singular knots in a smooth three-dimensional manifold, which is shown to great surprise to be a Kahler manifold. Many books on algebraic topology are written much too formally, and this makes the subject difficult to learn for students or maybe physicists who need insight, and not just functorial constructions, in order to learn or apply the subject. Fiber bundles are now ubiquitous in differential topology, algebraic topology, differential geometry, and algebraic geometry, and have also found a place in theoretical physics, thanks to the success of gauge field theories. amzp.siterank.org /us/cat/books/1100101687/6   (3240 words)

Comparison of topologies -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06) The finest topology on X is the (Click link for more info and facts about discrete topology) discrete topology. The coarsest topology on X is the (Click link for more info and facts about trivial topology) trivial topology. See (Click link for more info and facts about topologies on the set of operators on a Hilbert space) topologies on the set of operators on a Hilbert space for some intricate relationships. www.absoluteastronomy.com /encyclopedia/c/co/comparison_of_topologies.htm   (366 words)

Base (topology) In topology, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. The order topology is usually defined as the topology generated by a collection of open-interval-like sets. The metric topology is usually defined as the topology generated by a collection of open balls. news-server.org /b/ba/base__topology_.html   (691 words)

PlanetMath: order topology   (Site not responding. Last check: 2007-11-06) The subspace topology is always finer than the induced order topology, but they are not in general the same. Cross-references: contain, open set, open, singleton, finer, subspace, subspace topology, ordering, order, induced, subset, standard topologies, open intervals, basis, equivalent, open rays, subbasis, topology, linearly ordered set This is version 4 of order topology, born on 2002-01-06, modified 2004-12-07. planetmath.org /encyclopedia/OrderTopology.html   (159 words)

Math 101   (Site not responding. Last check: 2007-11-06) Tuesday, 9/23:Translation of "ordered pair" and "function" into set theory, definition of topology, example of the finite complement topology. Definition of one topology being finer than another. We proved that the topology on R x R generated by the basis of discs, and the topology generated by the basis of rectangles, were the same. www.math.harvard.edu /~jessica/math101/retro_syllabus.html   (350 words)

New Page 10   (Site not responding. Last check: 2007-11-06) A strategic topology on higher order belief types says that two types are close if they behave in similar ways in similar situations. An upper (lower) strategic topology on types is the finest topology generating upper (lower) hemicontinuity of strategic outcomes. We show that the upper strategic topology is equivalent to the product topology and the lower strategic topology is strictly finer. cowles.econ.yale.edu /conferences/uncertainty/morris-a.htm   (81 words)

Topological space Article, Topologicalspace Information   (Site not responding. Last check: 2007-11-06) The Zariski topology is a purely algebraically defined topologyon the spectrum of a ring or an algebraic variety. Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particularsequence of functions converges to the zero function. Any infinite set can be given the cofinite topology in whichthe open sets are the empty set and the sets whose complement is finite. www.anoca.org /spaces/set/topological_space.html   (2012 words)

People Finer   (Site not responding. Last check: 2007-11-06) Finer topology 1: s on ''X'', we say that andtau;1 is a '''finer ''' (alt. Jem Finer 1: MacGowan (with whom he co-wrote several songs) Finer was the most prolific composer for the band. Alabaster 30: The finer kinds of alabaster are largely employed as an orn www.lottery-news.net /dust30133-people_finer.html   (500 words)

Topology Course Lecture Notes , the topology for X induced by the metric d, is defined by agreeing that G shall be declared as open whenever each x in G is contained in an open ball entirely in G, i.e. That form of definition is useless in the absence of a properly defined 'distance' function but, fortunately, it is equivalent to the demand that the preimage of each open subset of the target metric space shall be open in the domain. 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)

Etale Cohomology. (PMS-33) Review and price   (Site not responding. Last check: 2007-11-06) This book is a rigorous overview of an approach to the study of schemes that uses a generalization of the complex topology called the etale topology. That the etale topology is a generalization of the complex topology is easier to see in the context of varieties rather than schemes as is done in this book. When reading this section, it is best to think of the fundamental group from ordinary topology in terms of the universal covering space instead of simple connectedness as it is the former concept that is employed to define the fundamental group of a scheme. www.wi-fitechnology.com /Wi-Fi-Products-0691082383.html   (1171 words)

Encyclopedia: Box topology In topology, the cartesian product of topological spaces is can be topologized in several ways. The canonical way of doing it is using the product topology, because it fits rather nicely with the categorical notion of a product. Another way to do it is with the box topology. www.nationmaster.com /encyclopedia/Box-topology   (594 words)

Finer Topology Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-11-06) Looking For finer topology - Find finer topology and more at Lycos Search. Find finer topology - Your relevant result is a click away! Look for finer topology - Find finer topology at one of the best sites the Internet has to offer! www.karr.net /encyclopedia/Finer_topology   (633 words)

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