Image with unknown copyright status removed Image line network.gif frame Image showing line network layout A linear bus topology is a network topology consisting of a main run of cable with a terminator at each end. All nodes file server, workstations, and peripherals are connected to the linear cable. Ethernet and LocalTalk networks use a linear bus topology. Advantages of a linear bus topology Easy to connect a computer or peripheral to a linear bus. Requires less cable length than a star topology . Disadvantages of a linear bus topology Entire network shuts down if there is a break in the main cable. Terminators are required at both ends of the backbone cable. Difficult to identify the problem if the entire network shuts down. Not meant to be used as a stand alone solution in a large building. External links http fcit.usf.edu network chap5 chap5.htm Category Network topology compu network stub id Topologi runtut ... 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
Infobox Writing system name Linear A type Undeciphered typedesc likely Syllabic and Ideographic languages Eteocretan language Eteocretan unknown time Possibly from MM IB to LM IIIA differentiated to Linear B and Linear Cypriot iso15924 Lina Image Linear A tablets filt.jpg thumb 237px right Linear A incised on tablets found in Akrotiri, Santorini . Linear A is one of two scripts used in ancient Crete before Mycenaean Greek language Mycenaean Greek Linear B Cretan hieroglyphs is the second script. In Minoan Civilization Minoan times, before the Mycenaean Greek dominion, Linear A was the official ... scripts were discovered and named by Arthur Evans . In 1952, Michael Ventris discovered that Linear ... of Linear B, although many points remain to be clarified. By contrast, Linear A has not been deciphered since the language of Linear A has not been discovered. Though the two scripts Linear A and B share some of the same symbols, using the syllable s associated with Linear B in Linear A writings produces ... 1450 BC. Linear A seems to have been used as a complete syllabary around 1900 1800 BC, although several signs appear earlier as mason marks. It is possible that the Trojan script Trojan Linear A scripts ... Linear A vase filt.jpg thumb right 180px Linear A incised on a vase, also found in Akrotiri. As the Minoan ... is correct. The simplest approach to decipherment may be to presume that the values of Linear A match more or less the values given to the fully transliterated Linear B script, used for Mycenean ... has a comprehensive list of known texts written in Linear A. ref This point of view has been ... in both Linear A and B therefore, 12 signs have the same values in both syllabaries DA, I, JA, KI .... This ambiguity represents the current state of understanding of the language of Linear A the known ... A The decipherment of Cretan inscriptions in Linear A in 1957 stating that Linear A contains Greek ... languages of Cretan inscriptions in Linear A , suggesting that the language of the Hagia Triada tablets ... more details
Unreferenced date December 2009 In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair , two vector space s with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one. Definition Given a dual pair math X, Y, langle , rangle math , a dual topology on math X math is a locally convex topology math tau math so that math X, tau simeq Y. math That is the continuous dual of math X, tau math is equal to math Y math up to linear isomorphism . Properties Theorem by George Mackey Mackey Given a dual pair, the bounded set topological vector space bounded set s under any dual topology are identical. Under any dual topology the same sets are barrelled set barrelled . Characterization of dual topologies The Mackey Arens theorem , named after George Mackey and Richard Friedrich Arens Richard Arens , characterizes all possible dual topologies on a locally convex space s. The theorem shows that the coarser topology coarsest dual topology is the weak topology , the topology of uniform convergence on all finite subsets of math X math , and the finer topology finest topology is the Mackey topology , the topology of uniform convergence on all weakly compact subsets of math ... and math X math its continuous dual then math tau math is a dual topology on math X math if and only if it is a topology of uniform convergence on a family of absolutely convex and weak topology weakly compact subsets of math X math DEFAULTSORT Dual Topology Category Topology of function spaces ... more details
Piecewise linear may refer to Piecewise linear function Piecewise linear manifold Piecewise linear continuation Collapse topology A polygonal chain or piecewise linear curve mathdab Long comment to avoid being listed on short pages ... more details
In functional analysis and related areas of mathematics , the Mackey topology , named after George Mackey , is the finer topology finest topology for a topological vector space which still preserves the continuous dual . In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology. The Mackey topology is the opposite of the weak topology , which is the coarser topology coarsest topology on a topological vector space which preserves the continuity of all linear functions in the continuous dual. The Mackey Arens theorem states that all possible dual topology dual topologies are finer than the weak topology and coarser than the Mackey topology. Definition Given a dual pair math X,X math with math X math a topological vector space and math X math its continuous dual the Mackey topology math tau X,X math is a polar topology defined on math X math by using the set of all absolutely convex and weak topology weakly compact sets in math X math . Examples Every metrisable locally convex space math X, tau math with continuous dual math X math carries the Mackey topology, that is math tau tau X, X math , or to put it more succinctly every Mackey space carries the Mackey topology Every Fr chet space math X, tau math carries the Mackey topology and the topology coincides with the strong topology , that is math tau tau X, X beta X, X math See also polar topology weak topology strong topology References springer id M m062080 title Mackey topology author A.I. Shtern cite journal last Mackey first G.W. authorlink George Mackey title On convex topological linear spaces journal Trans. Amer. Math. Soc. volume 60 year 1946 pages 519 537 doi 10.2307 1990352 issue 3 publisher Transactions of the American Mathematical Society, Vol. 60, No. 3 jstor 1990352 cite book last Bourbaki first Nicolas authorlink Nicolas Bourbaki title Topological vector spaces series Elements of mathematics publisher Addison Wesley year 1977 cite book last ... more details
incomplete date August 2009 In mathematics , general topology or point set topology is the branch of topology ... from other branches of topology in that the topological spaces may be very general, and do not have to be at all similar to manifold s. General topology provides the most general framework where fundamental concepts of topology such as open closed sets, continuity, interior exterior boundary points, and limit points could be defined. Definition Main Topological Space A topology is a pair X , .... History General topology grew out of a number of areas, most importantly the following the detailed study of subsets of the real line once known as the topology of point sets , this usage is now obsolete the introduction of the manifold concept the study of metric space s, esp. normed linear space s, in the early days of functional analysis . General topology assumed its present form around 1940 ..., it is in general topology that basic notions are defined and theorems about them proved. This includes the following open set open and closed set s interior topology interior and closure topology closure neighbourhood topology neighbourhood and closeness topology closeness compact space compactness and connected space connectedness continuous function topology continuous function mathematics ... branches of mathematics. Set theoretic topology examines such questions when they have substantial relations to set theory , as is often the case. Other main branches of topology are algebraic topology , geometric topology , and differential topology . As the name implies, general topology provides the common foundation for these areas. An important variant of general topology is pointless topology , which, rather than using sets of points as its foundation, builds up topological concepts ... and locales . See also List of examples in general topology Glossary of general topology for detailed definitions List of general topology topics for related articles Category of topological spaces ... more details
In mathematics , the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty , on the set of all closed subgroup s of a locally compact group G . The intuitive idea may be seen in the case of the set of all lattice group lattices in a Euclidean space E . There these are only certain of the closed subgroups others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. This topology can be derived from the Vietoris topology construction, a topological structure on all non empty subsets of a space. More precisely, it is an adaptation of the Fell topology construction, which itself derives from the Vietoris topology concept. References Claude Chabauty, Limite d ensembles et g om trie des nombres . Bulletin de la Soci t Math matique de France, 78 1950 , p. 143 151 Category Topological groups ... more details
maps see below. The weak topology weak convergence in normed linear space links to this heading ... with the topology of pointwise convergence of linear functionals. Other properties By definition, the weak ...dablink This article discusses the weak topology on a normed vector space. For the weak topology induced by a family of maps see initial topology . For the weak topology generated by a cover of a space see coherent topology . In mathematics , weak topology is an alternative term for initial topology . The term is most commonly used for the initial topology of a topological vector space such as a normed ... respectively, compact, etc. with respect to the weak topology. Likewise, functions are sometimes called ..., derivative differentiable , analytic function analytic , etc. with respect to the weak topology ... space topology such that addition, multiplication, and division are continuity topology ... is a K vector space equipped with a topological space topology so that vector addition and scalar multiplication are continuous. We may define a possibly different topology on X using the continuous or topological dual space X sup sup . The topological dual space consists of all linear operator linear functions from X into the base field K which are continuous function topology continuous with respect to the given topology. The weak topology on X is the initial topology with respect to X sup sup . In other words, it is the comparison of topologies coarsest topology the topology with the fewest ... topology from the original topology on X , the original topology is often called the strong topology . A subbase for the weak topology is the collection of sets of the form &phi sup 1 sup U where ... is open in the weak topology if and only if it can be written as a union of possibly infinitely many ..., if F is a subset of the algebraic dual space , then the initial topology of X with respect to F , denoted by &sigma X , F , is the weak topology with respect to F . If one takes F to be the whole ... more details
In mathematics , particularly in the area of functional analysis and topological vector space s, the vague topology is an example of the weak topology weak topology which arises in the study of measure theory measures on locally compact Hausdorff space s. Let X be a locally compact Hausdorff space . Let M X be the space of complex numbers complex Radon measure s on X , and C sub 0 sub X sup sup denote the dual of C sub 0 sub X , the Banach space of complex continuous function s on X vanish at infinity vanishing at infinity equipped with the uniform norm . By the Riesz representation theorem M X is isometry isometric to C sub 0 sub X sup sup . The isometry maps a measure &mu to a linear functional math I mu f int X f , d mu. math The vague topology is the Weak topology weak topology on C sub 0 sub X sup sup . The corresponding topology on M X induced by the isometry from C sub 0 sub X sup sup is also called the vague topology on M X . Thus, in particular, one may refer to vague convergence of measure &mu sub n sub &rarr &mu . One application of this is to probability theory for example, the central limit theorem is essentially a statement that if &mu sub n sub are the probability measure s for certain sums of independent random variables , then &mu sub n sub converge weakly to a normal distribution , i.e. the measure &mu sub n sub is approximately normal for large n . References citation author Dieudonn , Jean authorlink Jean Dieudonn chapter 13.4. The vague topology title Treatise on analysis volume II publisher Academic Press year 1970 . G.B. Folland, Real Analysis Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999. PlanetMath attribution id 7212 title Weak topology of the space of Radon measures Category Real analysis Category Topology of function spaces ... more details
limit together with the initial topology determined by the canonical morphisms. The weak topology on a locally convex space is the initial topology with respect to the continuous linear form s of its ...In general topology and related areas of mathematics , the initial topology or weak topology or limit topology or projective topology on a Set mathematics set math X math , with respect to a family of functions on math X math , is the coarsest topology on X which makes those functions continuous function topology continuous . The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these. The duality mathematics dual construction is called the final topology . Definition ... math f i X to Y i math the initial topology &tau on math X math is the coarsest topology on X such that each math f i X, tau to Y i math is continuous function topology continuous . Explicitly, the initial topology may be described as the topology subbase generated by sets of the form math f i 1 ... topology. The subspace topology is the initial topology on the subspace with respect to the inclusion map . The product topology is the initial topology with respect to the family of projection map ... topology on X with respect to the functions id sub X sub X &rarr X , &tau sub i sub is the supremum ... topology &tau is the topology generated by the union set theory union of the topologies &tau sub i sub . A topological space is completely regular if and only if it has the initial topology with respect ... X has the initial topology with respect to the family of continuous functions from X to the Sierpi ski space . Properties Characteristic property The initial topology on X can be characterized by the following ... 01.png center Characteristic property of the initial topology Evaluation By the universal property of the product topology we know that any family of continuous maps f sub i sub X &rarr Y sub i ... more details
In functional analysis and related areas of mathematics a polar topology , topology of math mathcal A math convergence or topology of uniform convergence on the sets of math mathcal A math is a method to define locally convex topology locally convex topologies on the vector space s of a dual pair . Definition Given a dual pair math X,Y, langle , rangle math and a family math mathcal A math of Set mathematics sets in math X math such that for all math A math in math mathcal A math the polar set math A 0 math is an absorbent set absorbent subset of math Y math , the polar topology on math Y math is defined by a family of semi norm s math p A A in mathcal A math . For each math A math in math mathcal A math we define math p A y sup vert langle x , y rangle vert x in A math . The semi norm math p A y math is the gauge mathematics gauge of the polar set math A 0 math . Examples a dual topology is a polar topology the converse is not necessarily true a locally convex topology is the polar topology defined by the family of equicontinuous sets of the dual space , that is the sets of all continuous linear form s which are equicontinuous Using the family of all finite sets in math X math we get the coarsest polar topology math sigma Y,X math on math Y math . math sigma Y,X math is identical to the weak topology . Using the family of all sets in math X math where the polar set is absorbent, we get the finest polar topology math beta Y,X math on math Y math Notes A polar topology is sometimes called topology of uniform convergence on the sets of math mathcal A math because given a dual pair math X,Y, langle , rangle math and a polar topology math tau math on math Y math defined by the gauges of the polar sets math A 0 math , a sequence math y n math in math Y, tau math converges to math y math if and only if for all semi norms math p A math math lim n to infty p A y n y lim n to infty ... with respect to math x in A math . Unreferenced date March 2008 Category Topology of function ... more details
Digital topology deals with properties and features of two dimensional 2D or Three dimensional space three dimensional 3D digital images that correspond to topological properties e.g., connectedness or topological features e.g., Boundary topology boundaries of objects. Concepts and results of digital topology are used to specify and justify important low level image analysis algorithms, including algorithms .... History Digital topology was first studied in the late 1960s by the computer image analysis researcher ... and developing the field. The term digital topology was itself invented by Rosenfeld, who used it in a 1973 publication for the first time. A related work called the grid cell topology appeared ... topology . Rosenfeld et al. proposed digital connectivity such as 4 connectivity and 8 connectivity ... grid cell topology to 3D and high dimensions. He also proposed 2008 a more general axiomatic theory ... . It is the Alexandrov topology. The book of 2008 contains new definitions of topological balls ... and computer vision. Basic results A basic early result in digital topology says that 2D binary ... use corresponds to open or closed sets in the 2D grid cell topology , and the result generalizes ... topology . Grid cell topology also applies to multilevel e.g., color 2D or 3D images, for example ... and Rosenfeld, 2004 . Digital topology is highly related to combinatorial topology . The main differences between them are 1 digital topology mainly studies digital objects that are formed by grid cells, clarify reason How does that differ? date October 2011 and 2 digital topology also deals with non .... It usually means a piecewise linear manifold made by simplicial complexes . A digital manifold ... topology Computational geometry Computational topology Topological data analysis Topology Discrete ... of Digital Discrete Geometry and Topology publisher SP Computing year 2004 isbn 0 9755122 1 8 cite book ... Category Digital topology fa ... 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
. See also lower limit topology long line topologyLinear continuum Notes references References ...one source date October 2011 In mathematics , an order topology is a certain topology that can be defined on any totally ordered set . It is a natural generalization of the topology of the real numbers ... generate nice non Hausdorff counter examples If X is a totally ordered set, the order topology ... a,b x mid a x b math together with the above rays form a base topology base for the order topology ... intervals and rays. The order topology makes X into a completely normal space completely normal ... topology If Y is a subset of X , then Y inherits a total order from X . Y therefore has an order topology, the induced order topology . As a subset of X , Y also has a subspace topology . The subspace topology is always at least as finer topology fine as the induced order topology, but they are not in general ... numbers rationals . Under the subspace topology, the singleton set 1 is open in Y , but under the induced order topology, any open set containing 1 must contain all but finitely many members of the space. An example of a subspace of a linearly ordered space whose topology is not an order topology Though the subspace topology of Y 1 &cup 1 n sub n &isin N sub in the section above is shown to be not generated by the induced order on Y , it is nonetheless an order topology on Y indeed, in the subspace topology every point is isolated i.e., singleton y is open in Y for every y in Y , so the subspace topology is the discrete topology on Y the topology in which every subset of Y is an open set , and the discrete topology on any set is an order topology. To define a total order on Y that generates the discrete topology on Y , simply modify the induced order on Y by defining 1 to be the greatest ... call it say sub 1 sub we have 1 n sub 1 sub 1 for all n &isin N . Then, in the order topology on Y ... ordered topological space X such that no total order on Z generates the subspace topology ... 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
, the entire network will be down. Linear bus The type of network topology in which all of the nodes ... the same fashion as the physical linear bus topology i.e., all nodes share a common transmission ... 50 ohm resistor. The linear bus topology is sometimes considered to be a special case of the distributed ... upon the physical star topology connected in a linear fashion i.e., daisy chained with no central ... of 1 would be classified as a physical lineartopology. The branching factor, f, is independent ... and ring. A lineartopology puts a two way link between one computer and the next. However, this was expensive ... a linear or ring topology is used to connect systems in multiple directions. A multi dimensional ... 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 ... more details
Algorithmic topology , or computational topology , is a subfield of topology with an overlap with areas of computer science , in particular computational geometry and computational complexity theory . A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithm s for solving topological problems, or using topological methods to solve algorithmic problems from other fields. Major algorithms by subject area Algorithmic 3 manifold theory A large family of algorithms concerning 3 manifold s revolve around normal surface theory, which is a phrase that encompasses several techniques to turn problems in 3 manifold theory into integer linear programming problems. Rubinstein and Thompson s 3 sphere recognition algorithm . This is an algorithm that takes as input ... . ref B. Burton. Introducing Regina, the 3 manifold topology software, Experimental Mathematics 13 ... word problem, Geometry and Topology 6 2002 1 26 ref At present the JSJ decomposition has ... classification of 3 manifolds can be done algorithmically. ref S.Matveev, Algorithmic topology and the classification ... a roughly linear run time in the number of crossings in the diagram, and low memory profile. The algorithm ... bound 4 manifolds. Journal of Topology 2008 1 3 703 745 ref S. Schleimer has an algorithm which ... although it is not widely considered implementable. See also Computational geometry Digital topology ... workshops WorkshopInfo 381 show workshop Workshop on Application of Topology in Science and Engineering http comptopfs.stanford.edu Computational Topology at Stanford University Books cite book author ... Afra J. Zomorodian title Topology for Computing url http books.google.com books?id oKEGGMgnWKcC publisher ... Topology An Introduction , Herbert Edelsbrunner, John L. Harer, AMS Bookstore, 2010, ISBN 978 0 8218 4925 5 DEFAULTSORT Computational Topology Category Computational topology Category Applied mathematics Category Computational complexity theory Category Computational science Topology ru ... 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 ... of all open intervals. In contrast to a basis linear algebra basis of a vector space in linear 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 ... 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
For other uses of triangulation in mathematics Triangulation disambiguation In mathematics , topology ... homology and cohomology theories. Piecewise linear structures For topological manifold s, there is a slightly stronger notion of triangulation a piecewise linear triangulation sometimes just called ... linear sphere. The link of a simplex s in a simplicial complex K is a subcomplex of K consisting ... simplex in K . For instance, in a two dimensional piecewise linear manifold formed by a set ... 3 fold Suspension topology suspension of the Poincar sphere is a topological manifold homeomorphic to the n sphere with a triangulation that is not piecewise linear it has a simplex whose ... manifold that is not homeomorphic to a sphere. The question of which manifolds have piecewise linear triangulations has led to much research in topology. Differentiable manifold s Stewart Cairns, harvs ... and Robert Hardt admit a piecewise linear triangulation. Topological manifold s of dimensions 2 and 3 are always triangulable by an Hauptvermutung essentially unique triangulation up to piecewise linear ... have an infinite number of triangulations, all piecewise linear inequivalent. In dimension greater ... it is known that some do not have piecewise linear manifold piecewise linear triangulations see ... Skeleton topology skeletons of Whitney triangulations are exactly the Neighbourhood graph theory ... last Milnor first John W. title Collected Works Vol. III, Differential Topology publisher American ... Topology, 1900 1960 publisher Birkh user year 1989 isbn 081763388X citation last Jost first J ... Edwin E. Moise last Moise first E. title Geometric Topology in Dimensions 2 and 3 publisher ... Differential Topology, revised edition series Annals of Mathematics Studies 54 publisher Princeton ... Thurston title Three Dimensional Geometry and Topology, Vol. I publisher Princeton University Press ... 164 doi 10.1016 0095 8956 92 90015 P Category Topology Category Algebraic topology Category Geometric ... 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
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
In algebraic geometry , the Zariski topology is a particular topology chosen for algebraic variety algebraic ... and took a place of particular importance in the field around 1950. The more subtle tale topology is a refinement of the Zariski topology discovered by Grothendieck in the 1960s that reflects the geometry ... prior to the Grothendieck revolution of the late 1950s and 1960s the Zariski topology was defined ... varieties see algebraic variety Formal definitions the Algebraic variety definitions the Zariski topology ... varieties First we define the topology on affine spaces math mathbb A n, math which as sets are just n dimensional vector spaces over k . The topology is defined by specifying its closed, rather than ... that these sets form the closed sets of a topology equivalently, their complements, denoted D S and called principal open sets , form the topology itself . This is the Zariski topology on math mathbb A n. math If X is an affine algebraic set irreducible or not then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some math mathbb A n. math Equivalently ... of the previous one, defines the Zariski topology on any affine variety. Projective ... polynomials, define a topology on math mathbb P n. math As above the complements of these sets are denoted D S , or, if confusion is likely to result, D&prime S . The projective Zariski topology ..., by taking the subspace topology. Similarly, it may be shown that this topology is defined intrinsically ... A very useful fact about these topologies is that we may exhibit a base topology basis for them consisting ... ring is Noetherian. As a consequence, affine or projective spaces with the Zariski topology ... function topology continuous in the Zariski topology. In fact, the Zariski topology is the weakest topology ... polynomial is irreducible if and only if it is linear, of the form t &minus a , for some element ... s, the picture becomes more complicated because of the existence of non linear irreducible polynomials ... more details
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