PathTopology Category Topology Category Homotopy theory de Weg Mathematik fr Chemin topologie ko ...Image Path.svg thumb The points traced by a path from A to B in R . However, different paths can trace the same set of points. In mathematics , a path in a topological space X is a continuous topology continuous map f from the unit interval I 0,1 to X f I &rarr X . The initial point of the path is f 0 and the terminal point is f 1 . One often speaks of a path from x to y where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which looks like a curve ... sup 2 sup represent two different paths from 0 to 1 on the real line. A Loop topology loop in a space X based at x X is a path from x to x . A loop may be equally well regarded as a map f I X with f 0 ... a path connecting any two points is said to be path connected space path connected . Any space may be broken up into a set of path connected component s. The set of path connected components ... s, which are important in homotopy theory . If X is a topological space with basepoint x sub 0 sub , then a path ... topology called homotopy theory . A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed. Specifically, a homotopy of paths, or path homotopy ... precisely path homotopic , to distinguish between the relation defined on all continuous functions between ... class of a path f under this relation is called the homotopy class of f , often denoted f . Path composition One can compose paths in a topological space in an obvious manner. Suppose f is a path from x to y and g is a path from y to z . The path fg is defined as the path obtained by first traversing ... s leq 1. end cases math Clearly path composition is only defined when the terminal point of f coincides with the initial point of g . If one considers all loops based at a point x sub 0 sub , then path composition is a binary operation . Path composition, whenever defined, is not associative due to the difference ... more details
Topology broadcast based on reverse path forwarding TBRPF is a link state routing protocol for wireless mesh network s. The obvious design for a wireless link state protocol such as the optimized link state routing protocol transmits large amounts of routing data, and this limits the utility of a link state protocol when the network is made of moving nodes. The number and size of the routing transmissions make the network unusable for any but the smallest networks. The conventional solution is to use a distance vector routing protocol such as AODV , which usually transmits no data about routing. However, distance vector routing requires more time to establish a connection Citation needed date April 2009 , and the routes are less optimized than a link state router Citation needed date April 2009 . TBRPF transmits only the differences between the previous network state and the current network state. Therefore, routing messages are smaller, and can therefore be sent more frequently. This means that nodes routing tables are more up to date. TBRPF is controlled under a United States patent law US patent filed in December 2000 and assigned to SRI International Patent ID 6845091, issued January 18, 2005 . Links RFC 3684 Topology Dissemination Based on Reverse Path Forwarding TBRPF http www.packethop.com Packethop Inc. website http www.sri.com esd projects tbrpf References B. Bellur, and R.G. Ogier. 1999. A Reliable, Efficient Topology Broadcast Protocol for Dynamic Networks, Proc. IEEE INFOCOMM 99, pp. 178 186. R.G. Ogier, M.G. Lewis, F.L. Templin, and B. Bellur. 2002. Topology Broadcast based on Reverse Path Forwarding TBRPF , RFC 3684. Category Wireless networking Category Ad hoc routing protocols ... more details
one surface and one edge. Such shapes are an object of study in topology. Topology from the Greek ... or analysis situs Greek Latin for picking apart of place . This later acquired the modern name of topology Specify . By the middle of the 20th century, topology had become an important area of study within mathematics. The word topology is used both for the mathematical discipline and for a family ... object of topology. Of particular importance are homeomorphism s , which can be defined as continuous function s with a continuous inverse function inverse . Topology includes many subfields. The most basic and traditional division within topology is General topology point set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces basic examples include compactness and connectedness algebraic topology , which generally tries ... mathematics homology and geometric topology , which primarily studies manifold s and their embeddings placements in other manifolds. Some of the most active areas, such as low dimensional topology and graph ... , the simplest non trivial knot See also topology glossary for definitions of some of the terms used in topology and topological space for a more technical treatment of the subject. History Image Konigsberg .... Topology began with the investigation of certain questions in geometry. Leonhard Euler s 1736 ... academic treatises in modern topology. The term Topologie was introduced in German in 1847 by Johann ... years in correspondence before its first appearance in print. Topology, its English form, was first ... topologist in the sense of a specialist in topology was used in 1905 in the magazine The Spectator ... definition of topology. Modern topology depends strongly on the ideas of set theory , developed ... of homotopy and homology mathematics homology , which are now considered part of algebraic topology ..., see point set topology and algebraic topology . Elementary introduction Topology, as a branch ... more details
About uses of path and pathway the acronym PATHPATH disambiguation PATH wiktionary path pathway TOC right Path , pathway or PATH may refer to Path Course navigation , the intended path of a vehicle over the surface of the Earth Trail , hiking trail , footpath , or bridle path See also Track disambiguation Footpath disambiguation Shining Path , Maoist guerrilla insurgent organization in Peru Sidewalk running along the edge of a road, in some varieties of English Bicycle path or bikeway way Golden Path Dune , a metaphysical theme from Frank Herbert s Dune novels Path Vol.2 is a 2000 single by Apocalyptica from their album Cult Mathematics Path graph theory , a sequence of vertices of a graph Pathtopology , a continuous function Computing Path computing , in computer file systems, the human readable address of a resource PATH variable , an environment variable specifying a list of directories where executable programs are located Path social network , a social networking enabled photo sharing and messaging service Clipping path , a computer image outlining option to remove background and create transparency Control flow path, a possible execution sequence in a program often depicted as a sequence of edges in a control flow graph The st connectivity problem is sometimes known as the path problem. Pathway Biology Genetic pathway , a group of genes interacting to form an aggregate biological function Metabolic pathway , a series of chemical reactions within a cell Signal transduction Signalling pathway , a series of interactions e.g. from cell receptors to affect gene expression. Neural pathway , a neural tract connecting one part of the nervous system with another Dopaminergic ... Path , various businesses founded and originally run by the Path Brothers of France PATH disambiguation , disambiguation page for the acronym PATH The Path disambiguation disambiguation cs Path da Sti de PATH fr Path ko nl Pad pt Caminho simple Path ... more details
The Path may refer to The Path album The Path album , a 2003 studio album by Show Of Hands The Path book The Path book , collection of short essayes by Konosuke Matsushita The Path comics The Path comics , an American comic book series by CrossGen Entertainment The Path video game The Path video game , a psychological horror art PC game See also Path disambiguation disambig fr The Path it The Path ... more details
About the acronym PATH other uses of pathPath disambiguation PathPATH may refer to Port Authority Trans Hudson , a subway system linking Manhattan, New York with locations in northern New Jersey PATH Atlanta , trail building organization Georgia, USA PATH Toronto , a network of underground pedestrian tunnels in Toronto, Ontario, Canada Partners for Advanced Transit and Highways , a research organization operated by the University of California The Performance Assessment Tool For Quality Improvement In Hospitals , a performance assessment system designed by the World Health Organization to support hospitals in defining quality improvement strategies, questioning their own results and translating them into actions for improvement. Positive Alternatives to Homosexuality , a coalition of ex gay organizations Program for Appropriate Technology in Health , an international, nonprofit organization based in Seattle, Washington, USA Projects for Assistance in Transition from Homelessness , to support service delivery to individuals with serious mental illnesses who are homeless or at risk of becoming homeless Potomac Appalachian Transmission Highline , proposed electrical line PATH variable , a computer operating system environment variable specifying a list of directories where executable programs are located disambig ... more details
Unreferenced date December 2006 orphan date November 2009 In topology , a hereditarily unicoherent , Connected space Path connectedness arcwise connected continuum topology continuum is called a dendroid. A continuum X is called hereditarily unicoherent if every subcontinuum of X is unicoherent . A locally connected dendroid is called a dendrite mathematics dendrite . DEFAULTSORT Dendroid Topology Category Continuum theory Topology stub ... more details
Infobox film name On the Path image On the Path.jpg image size caption director Jasmila bani producer Damir Ibrahimovic writer Jasmila bani starring Mirjana Karanovi music cinematography Christine A. Maier editing distributor released Film date 2010 2 18 60th Berlin International Film Festival Berlinale 2010 2 20 Bosnia and Herzegovina runtime country Bosnia and Herzegovina language Bosnian budget On the Path lang bs Na putu is a 2010 Bosnian and Herzegovinan drama film directed by Jasmila bani . Plot Luna and Amar are a young Bosniaks Bosnian couple living in Sarajevo. Both have traumatic memories from the Bosnian War of the 1990 s. Luna had seen her parents killed by an anti Muslim militia in Bijeljina , and had come to Sarajevo with her grandparents as a child refugee. Amar had served as a soldier in the war and lost his brother. At present, however, they have apparently built up a successful life she as an air hostess with B&H Airlines , he as an air traffic controller at the Sarajevo International Airport . When she comes back from a flight they make love passionately and go to have a good time at a local nightclub. Though identifying as Islam in Bosnia and Herzegovina Muslim s in the context of Bosnia s ethnic set up, religion plays no part in their life. In fact, Amar drinks alcoholic drinks a bit too much which is forbidden by Islam and it is this which begins to put their relationship under strain. First of all, Amar loses his job for being drunk at work. Luna is very worried and has little hope of realizing her fragile dream of having a child with Amar. But her fears for their future increase when Amar takes on a well paid job in a Muslim community hours away from where they live. Only after quite some time has elapsed during which they have had no contact ... and Amar together on the path to a lifetime of happiness. Cast Zrinka Cvite i Leon Lu ev Mirjana ... accessdate 2011 01 01 ref References reflist External links imdb title 1156531 DEFAULTSORT On The Path ... more details
In mathematics , particularly topology , the K topology is a Topological space topology that one can impose on the set of all real numbers which has some interesting properties. Relative to the set of all real numbers carrying the standard topology , the set K 1 n n is a natural number is not Closed set closed since it doesn t contain its only limit point 0. Relative to the K topology however, the set K is automatically decreed to be closed by adding more Basis basis elements dn date April 2012 to the standard topology on R . Basically, the K topology on R is strictly finer than the standard topology on R . It is mostly useful for counterexamples in basic topology. Formal definition Let R be the set of all real numbers and let K 1 n n is a natural number . Generate a topology on R by taking as Basis topology basis all open intervals a , b and all sets of the form a , b K the set of all elements in a , b that are not in K . The Topological space topology generated is known as the K topology on R . Note that The sets described in the definition do form a basis they satisfy the conditions to be a basis . Properties and examples Throughout this section, T will denote the K topology and R , T will denote the set of all real numbers with the K topology as a topological space . 1. The topology T on R is strictly finer than the standard topology on R but not comparable with the lower limit topology on R 2. From the previous example, it follows that R , T is not Compact space compact 3 ... topological space . However, R , T is not Connected space path connected it has precisely two Locally connected space path components &minus &infin , 0 and 0, &infin 5. Note also that R , T is not Locally connected space locally path connected since its path components are not equal to its Locally ... space Hausdorff . See also Lower limit topology Natural topology Sequence Locally connected space Connected space References cite book author James Munkres year 1999 title Topology edition 2nd edition ... more details
Spacetime topology , the Topological space topological structure of spacetime , is a subject studied ... and the concepts of topology thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in physical cosmology . Types of topology There are two main types of topology for a spacetime math M math Manifold topology As with any manifold, a spacetime possesses a natural manifold topology. Here the open set s are the image of open sets in math mathbb R 4 math . Path or Zeeman topology Definition ref name Bombelli http www.phy.olemiss.edu 7Eluca Topics t top st.html Luca Bombelli website ref The topology math rho math in which a subset math E subset M math is open topology open if for every timelike curve math c math there is a set math O math in the manifold topology such that math E cap c O cap c math . It is the finest topology which induces the same topology as math M math does on timelike curves. Properties Strictly finer topology finer than the manifold topology. It is therefore Hausdorff space Hausdorff , Separable topology separable but not Locally compact space locally compact . A Base topology base for the topology is sets of the form math I p,U cup I p,U cup p math for some point math p in M ... structure Causal structure chronological past and future . Alexandrov topology The Alexandrov topology on spacetime, is the Comparison of topologies coarsest topology such that both math I E math and math I E math are open for all subsets math E subset M math . Here the Base topology base of open set s for the topology are sets of the form math I x cap I y math for some points math ,x,y in M math . This topology coincides with the manifold topology if and only if the manifold is Causality conditions ... topology on a partial order is usually taken to be the coarsest topology in which only the upper ... topology on spacetime would be the interval topology , but when Kronheimer and Penrose introduced ... more details
X is continuous function topology continuous . X is path connected and so connected space connected ...In topology , a topological space with the trivial topology is one where the only open set s are the empty set and the entire space. Such a space is sometimes called an indiscrete space , and its topology sometimes called an indiscrete topology . Intuitively, this has the consequence that all points ... zero . The trivial topology is the topology with the least possible number of open set s, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space X with more than 1 number one element and the trivial topology lacks a key desirable property it is not a T0 ... unusual&mdash include The only closed set s are the empty set and X . The only possible basis topology ... space . Not being Hausdorff, X is not an order topology , nor is it metrizable . X is, however ... countable , separable space separable and Lindel f space Lindel f . All subspace topology subspace s of X have the trivial topology. All quotient space s of X have the trivial topology Arbitrary product space product s of trivial topological spaces, with either the product topology or box topology , have the trivial topology. All sequence s in X limit mathematics converge to every point of X . In particular ... compact . The interior topology interior of every set except X is empty. The closure topology closure ... carrying the trivial topology are homeomorphic iff they have the same cardinality . In some sense the opposite of the trivial topology is the discrete topology , in which every subset is open. The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage topology entourage . Let Top be the category of topological spaces with continuous ... that puts the trivial topology on a given set, then G is adjoint functors right adjoint to F . The functor H Set &rarr Top that puts the discrete topology on a given set is adjoint functors ... more details
In topology and related areas of mathematics , a product space is the cartesian product of a family of topological space s equipped with a natural topology called the product topology . This topology differs from another, perhaps more obvious, topology called the box topology , which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is correct in that it makes the product space a product category theory categorical product of its factors, whereas the box topology is Comparison of topologies too fine this is the sense in which the product topology is natural . Definition Given X such that math ... p sub i sub X &rarr X sub i sub , the product topology on X is defined to be the coarsest topology i.e. the topology with the fewest open sets for which all the projections p sub i sub are continuous topology continuous . The product topology is sometimes called the Tychonoff topology . The open sets in the product topology are unions finite or infinite of sets of the form math prod U i ... many times. The product topology on X is the topology generated by sets of the form p sub i sub ... p sub i sub sup &minus 1 sup U form a subbase for the topology on X . A subset of X is open if and only ... a basis topology basis for the product topology using bases of the constituting spaces X sub i sub . A basis ... topology on X . In general, the box topology is finer topology finer than the product topology, but for finite products they coincide. Examples If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on R sup n sup . The Cantor set is homeomorphic to the product of countable countably ... topology. Several additional examples are given in the article on the initial topology . Properties ... if W is a subspace topology subspace of the product space whose projections down to all the X sub ... more details
Unreferenced date November 2009 A loop in mathematics , in a topological space X is a pathtopologypath f from the unit interval I 0,1 to X such that f 0 f 1 . In other words, it is a path whose initial point is equal to the terminal point. A loop may also be seen as a continuous map f from the unit circle S sup 1 sup into X , because S sup 1 sup may be regarded as a quotient space quotient of I under the identification 0 1. The set of all loops in X forms a space called the loop space of X . See also Free loop Loop group Loop space Loop algebra Fundamental group DEFAULTSORT Loop Topology Category Topology es Grupo fundamental Lazo fr Lacet math matiques nl Lus topologie pt Lacete ... more details
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 the final topology on the disjoint union topology disjoint union the topology arising from a normed vector space norm the strong operator topology the strong topology polar topology , which subsumes all topologies above. Note that a topology is stronger than a topology is a Comparison of topologies finer topology if contains all the open sets of . In algebraic geometry , it usually means the topology of an algebraic variety as complex manifold or subspace of complex projective space , as opposed to the Zariski topology which is rarely even a Hausdorff space . See also Weak topology mathdab Category Topology ... more details
In functional analysis and related areas of mathematics the strong topology is the finer topology finest polar topology , the topology with the most open set s, on a dual pair . The coarser topology coarsest polar topology is called weak topology polar topology weak topology . Definition Given a dual pair math X,Y, langle , rangle math the strong topology math beta Y, X math on math Y math is the polar topology defined by using the family of all sets in math X math where the polar set in math Y math is Absorption law absorbent . Examples Given a normed vector space math X math and its continuous dual math X math then math beta X , X math topology on math X math is identical to the topology induced by the operator norm . Conversely math beta X, X math topology on math X math is identical to the topology induced by the norm mathematics norm . Properties In barrelled space s the strong topology is identical to the Mackey topology . mathanalysis stub Category Topology of function spaces ... more details
In mathematics , the uniform topology on a space has several different meanings depending on the context In functional analysis, it sometimes refers to a polar topology on a topological vector space. In general topology, it is the topology carried by a uniform space . In real analysis, it is the topology of uniform convergence . Disambig ... more details
. Properties All cones are path connected space path connected since every point can be connected to the vertex ... t sub x , s x , 1&minus t s . The cone is used in algebraic topology precisely because it embeds a space as a subspace topology subspace of a contractible space. When X is compact space compact and Hausdorff ... fails when X is not compact or not Hausdorff, as generally the quotient topology on CX will be finer topology finer than the set of lines joining X to a point. Reduced cone If math X,x 0 math ... Top math on the category of topological spaces Top . See also Cone disambiguation Suspension topology Mapping cone Join topology References Allen Hatcher , http www.math.cornell.edu hatcher AT ATpage.html Algebraic topology. Cambridge University Press, Cambridge, 2002. xii 544 pp. ISBN 0 521 79160 X and ISBN 0 521 79540 0 planetmath reference id 3974 title Cone Category Topology Category Algebraic topology fr C ne topologie it Cono topologia pl Sto ek topologia ... more details
In mathematics, fine topology can refer to Fine topology potential theory The sense opposite to coarse topology , namely A term in comparison of topologies which specifies the partial order relation of a topological structure to other one s Final topology See also Discrete topology , the most fine topology possible on a given set mathdab ... more details
300px Diagram of different network topologies. Network topology is the layout pattern of interconnections ... url http www.atis.org glossary definition.aspx?id 3516 title network topology author ATIS committee ... be physical or logical. Physical topology refers to the physical design of a network including the devices, location and cable installation. Logical topology refers to how data is actually transferred in a network as opposed to its physical design. In general physical topology relates to a core network whereas logical topology relates to basic network. Topology can be understood as the shape or structure ... not necessarily mean that it represents a ring topology . Any particular network topology ... between nodes. The study of network topology uses graph theory . Distances between nodes, physical ... a physical topology and a logical topology. Any given node in the LAN has one or more links to one ... shape that may be used to describe the physical topology of the network. Likewise, the mapping of the data flow between the nodes in the network determines the logical topology of the network. The physical and logical topologies may or may not be identical in any particular network. Topology There are two ... layout used to link devices is called the physical topology of the network. This refers ... the nodes and the cabling. ref name Groth The physical topology of a network is determined by the capabilities ..., and the cost associated with cabling or telecommunications circuits. The logical topology, in contrast .... A network s logical topology is not necessarily the same as its physical topology. For example, the original twisted pair Ethernet using repeater hub s was a logical bus topology with a physical star topology layout. Token Ring is a logical ring topology, but is wired a physical star from the Media ... as those in the physical classifications of network topologies but describes the path that the data ... match the logical flow of data, hence the convention of using the terms logical topology and signal ... more details
Connected component Path connected component . Connected topology Connected A space is connected topology ... X , a loop topology loop at x in X or a loop in X with basepoint x is a path f in X , such that f ... on the entire space is identically 1. PathtopologyPath A Pathtopologypath in a space X is a continuous ... as topology . Although there is no absolute distinction between different areas of topology, the focus here is on general topology . The following definitions are also fundamental to algebraic topology , differential topology and geometric topology . See the article on topological space s for basic definitions and examples, and see the article on topology for a brief history and description of the subject ... contain specialised vocabulary within general topology or provide more detailed expositions of the definitions given below. The list of general topology topics and the list of examples in general topology will also be very helpful. Compact space Connected space Continuity topology Continuity Metric ... yes A Accessible See T1 space math T 1 math . Accumulation point See limit point . Alexandrov topology A space X has the Alexandrov topology or is finitely generated if arbitrary intersections of open ... the natural numbers to the natural numbers, with the topology of pointwise convergence see Baire space set theory . Base topology Base A collection B of open sets is a base topology base or basis for a topology math tau math if every open set in math tau math is a union of sets in math B math . The topology math tau math is the smallest topology on math X math containing math B math and is said to be generated by math B math . Basis topology Basis See Base topology Base . Borel algebra The Borel .... Boundary topology Boundary The boundary topology boundary or frontier of a set is the set s closure ... set in the topology induced on M by d . Note that the closed ball D x r might not be equal to the closure topology closure of the open ball B x r . Closed set A set is Closed set closed if its ... more details
In mathematics, coarse topology is a term in comparison of topologies which specifies the partial order relation of a topological structure to other one s . Specifically, it may refer to Initial topology , the most coarse topology in a certain category of topologies Trivial topology , the most coarse topology possible on a given set See also Weak topology , an example of topology coarser than the standard one Fine topology disambiguation mathdab ... more details
In mathematics , a base or basis B for a topological space X with topological space topology T is a collection ... . We say that the base generates the topology T . Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies ... properties of bases are The base elements cover topology cover X . Let B sub 1 sub , B sub ... either of these, then it is not a base for any topology on X . It is a subbase , however, as is any ... topology on X for which B is a base it is called the topology generated by B . This topology ... common way of defining topologies. A sufficient but not necessary condition for B to generate a topology ..., the collection of all open interval s in the real line forms a base for a topology on the real ... they are a base for the standard topology on the real number s. However, a base is not unique. Many bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational ... algebra , a base need not be maximal indeed, the only maximal base is the topology itself. In fact ... the topology. The smallest possible cardinality of a base is called the weight of the topological space ... of the forms , a and a , , where a is a real number. Then S is not a base for any topology on R . To show this, suppose it were. Then, for example, , 1 and 0, would be in the topology generated ... property fails, since no base element can fit inside this intersection. Given a base for a topology ... 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 ball s. A second countable space is one that has a countable base. The discrete topology has the Singleton ... element containing x and contained in U . A topology T sub 2 sub is topological space finer than a topology ... more details
Unreferenced date December 2009 In functional analysis , a branch of mathematics , the ultraweak topology , also called the weak topology , or weak operator topology or weak topology , on the set B H of bounded operator s on a Hilbert space is the weak topology weak topology obtained from the predual B sub sub H of B H , the trace class operators on H . In other words it is the weakest topology such that all elements of the predual are continuous when considered as functions on B H . Relation with the weak operator topology The ultraweak topology is similar to the weak operator topology. For example, on any norm bounded set the weak operator and ultraweak topologies are the same, and in particular the unit ball is compact in both topologies. The ultraweak topology is stronger than the weak operator topology. One problem with the weak operator topology is that the dual of B H with the weak operator topology is too small . The ultraweak topology fixes this problem the dual is the full predual B sub sub H of all trace class operators. In general the ultraweak topology is more useful than the weak operator topology, but it is more complicated to define, and the weak operator topology is often more apparently convenient. The ultraweak topology can be obtained from the weak operator topology as follows. If H sub 1 sub is a separable infinite dimensional Hilbert space then B H can be embedded in B H H sub 1 sub by tensoring with the identity map on H sub 1 sub . Then the restriction of the weak operator topology on B H H sub 1 sub is the ultraweak topology of B H . See also Topologies on the set of operators on a Hilbert space ultrastrong topology weak operator topology DEFAULTSORT Ultraweak Topology Category Topology of function spaces Category Von Neumann algebras ... more details
Unreferenced date December 2009 In functional analysis and related areas of mathematics the weak topology is the coarser topology coarsest polar topology , the topology with the fewest open set s, on a dual pair . The finer topology finest polar topology is called strong topology polar topology strong topology . Under the weak topology the Bounded set topological vector space bounded set s coincide with the relatively compact set s which leads to the important Bourbaki Alaoglu theorem . Definition Given a dual pair math X,Y, langle , rangle math the weak topology math sigma X,Y math is the weakest polar topology on math X math so that math X, sigma X,Y simeq Y math . That is the continuous dual of math X, sigma X,Y math is equal to math Y math up to isomorphism . The weak topology is constructed as follows For every math y math in math Y math on math X math we define a semi norm on math X math math p y X to mathbb R math with math p y x vert langle x , y rangle vert qquad x in X math This family of semi norms defines a locally convex topology on math X math . Examples Given a normed vector space math X math and its continuous dual math X math , math sigma X, X math is called the weak topology on math X math and math sigma X , X math the weak star topology weak topology on math X math DEFAULTSORT Weak Topology Polar Topology Category Topology of function spaces ... more details
Logical topology also referred to as signal topology is a network computing term used to describe the arrangement of devices on a network and how they communicate with one another. How devices are connected to the network through the actual cables that transmit data, or the physical structure of the network, is called the Network topology physical topology . Logical topologies are bound to network protocols and describe how data is moved across the network. There are attempts to study the logical topology of the Internet by network scientists such as Albert L szl Barab si . Category Network topology ar ... more details
Encyclopedia
top
