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
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
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
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 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
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 May 2011 In any domain of mathematics , a space has a natural topology if there is a topology on the space which is best adapted to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that the topology in question arises naturally or canonically see mathematical jargon in the given context. Note that in some cases multiple ... X , then the Order topology Induced order topology induced order topology , i.e. the order topology of the totally ordered Y , where this order is inherited from X , is coarser than the subspace topology of the order topology of X . Natural topology does quite often have a more specific meaning, at least given some prior contextual information the natural topology is a topology which makes a natural map or collection of maps Continuous function topology continuous . This is still imprecise, even ... property. However, there is often a finest topology finest or coarsest topology coarsest topology ... topology. The simplest cases which nevertheless cover many examples are the initial topology and the final topology Willard 1970 . The initial topology is the coarsest topology on a space X which makes a given collection of maps from X to topological spaces X sub i sub continuous. The final topology is the finest topology on a space X which makes a given collection of maps from topological spaces ... and quotient spaces. The natural topology on a subset of a topological space is the subspace topology . This is the coarsest topology which makes the inclusion map continuous. The natural topology on a quotient space quotient of a topological space is the quotient topology . This is the finest topology which makes the quotient map continuous. Other examples include the topology induced by the Helly metric . References cite book last Willard first Stephen title General Topology publisher Addison ... Mathematical structures Category Topologytopology stub ... 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
In functional analysis , the ultrastrong topology , or &sigma strong topology , or strongest topology on the set B H of bounded operator s on a Hilbert space is the topology defined by the family of seminorms math p omega x omega x x 1 2 math for positive elements math omega math of the predual math ... John title On a Certain Topology for Rings of Operators journal The Annals of Mathematics 2nd Ser ... 292 3A37 3A1 3C111 3AOACTFR 3E2.0.CO 3B2 S ref Relation with the strong operator topology The ultrastrong topology is similar to the strong operator topology. For example, on any norm bounded set the strong operator and ultrastrong topologies are the same. The ultrastrong topology is stronger than the strong operator topology. One problem with the strong operator topology is that the dual of B H with the strong operator topology is too small . The ultrastrong topology fixes this problem the dual is the full predual B sub sub H of all trace class operators. In general the ultrastrong topology is better than the strong operator topology, but is more complicated to define so people usually use the strong operator topology if they can get away with it. The ultrastrong topology can be obtained from the strong operator topology as follows. If H sub 1 sub is a separable infinite dimensional Hilbert ... sub 1 sub . Then the restriction of the strong operator topology on B H &otimes H sub 1 sub is the ultrastrong topology of B H . Equivalently, it is given by the family of seminorms math x mapsto left ... rp 68 The adjoint map is not continuous in the ultrastrong topology. There is another topology called the ultrastrong sup sup topology, which is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous. ref name TakesakiI rp 68 See also Topologies on the set of operators on a Hilbert space ultraweak topology strong operator topology References Reflist Category Topology of function spaces Category von Neumann algebras ... more details
Infobox Book name Counterexamples in Topology image image caption author Lynn Steen Lynn Arthur Steen ... Counterexamples in Topology 1970, 2nd ed. 1978 is a book on mathematics by topology topologist s Lynn ... a counterexample which exhibits one property but not the other. In Counterexamples in Topology , Steen ... , Minnesota in the summer of 1967, canvassed the field of topology for such counterexamples and compiled ... space which is not second countable space second countable is counterexample 3, the discrete topology ... of metrization theory and general topology see History of the separation axioms for more. List of mentioned counterexamples colbegin cols 2 finite set Finite discrete topology Countable discrete topology Uncountable discrete topology Indiscrete topology Partition topology Odd even topology Deleted integer topology Particular point topology Finite particular point topology Particular point topology Countable particular point topology Particular point topology Uncountable particular point topology Sierpinski space , see also particular point topology Closed extension topology Finite excluded point topology Countable excluded point topology Uncountable excluded point topology Open extension topology Either or topology Finite complement topology on a countable space Finite complement topology on an uncountable space Countable complement topology Double pointed countable complement topology Compact complement topology Countable Fort space Uncountable Fort space Fortissimo space Arens Fort space Modified Fort space Euclidean space Euclidean topology Cantor set Rational number s Irrational ... topology One point compactification of the rationals Hilbert space Fr chet space Hilbert cube Order topology Open ordinal space 0, where Closed ordinal space 0, where Open ordinal space 0, Closed ordinal space 0, Uncountable discrete ordinal space Long line topology Long line Long line topology Extended long line An altered Long line topology long line Lexicographic order topology ... 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. Definition A topology is a pair X , consisting of a set mathematics ... intersection of open sets is an open set. X and the empty set are open sets. History General topology ... line once known as the topology of point sets , this usage is now obsolete the introduction ... of functional analysis . General topology assumed its present form around 1940. It captures, one might ... 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 function s limit of a sequence .... 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 ... 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 References Some standard books on general topology include Bourbaki cite Topologie G n rale cite cite General Topology cite ISBN 0 387 19374 X John L. Kelley cite General Topology cite ISBN 0 387 90125 6 James Munkres cite Topology cite ISBN 0 13 181629 2 Paul L. Shick cite Topology Point Set and Geometric cite ISBN 0 470 09605 5 Ryszard Engelking cite General Topology cite ISBN 3 88538 006 4 Citation last1 Steen first1 ... more details
In topology , a branch of mathematics , an extension topology is a topology structure topology placed ... of extension topology, described in the sections below. Extension topology Let X be a topological space and P a set disjoint from X. Consider in X P the topology whose open sets are of the form ... of P. For these reasons this topology is called the extension topology of X plus P, with which one extends to X P the open and the closed sets of X. Note that the subspace topology of X as a subset of X P is the original topology of X, while the subspace topology of P as a subset of X P is the discrete space discrete topology . Being Y a topological space and R a subset of Y, one might ask whether the extension topology of Y R plus R is the same as the original topology of Y, and the answer is in general no. Note the similitude of this extension topology construction ... topology Let X be a topological space and P a set disjoint from X. Consider in X P the topology ... set of X. For this reason this topology is called the open extension topology of X plus P, with which one extends to X P the open sets of X. Note that the subspace topology of X as a subset of X P is the original topology of X, while the subspace topology of P as a subset of X P is the discrete space discrete topology . Note that the closed sets of X .... Being Y a topological space and R a subset of Y, one might ask whether the extension topology of Y R plus R is the same as the original topology of Y, and the answer is in general no. Note that the open extension topology of X P is comparison of topologies smaller than the extension topology of X P. Being Z a set and p a point in Z, one obtains the excluded point topology construction by considering in Z the discrete space discrete topology and applying the open extension topology construction to Z p plus p. Closed extension topology Let X be a topological space and P ... 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
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
Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation stretching without tearing or gluing these properties are the topological invariants. Topology may also refer to Topology, the collection of open sets used to define a topological space Topology journal Topology journal , a mathematical journal, with an emphasis on subject areas related to topology and geometry Topology, a term used in architecture to describe spatial effects which cannot be described by topography, i.e., social, economical, spatial or phenomenological interactions Topology, a term used in cell biology to describe the Membrane topology specific orientation of transmembrane proteins . Topology electronics , a configuration of electronic components. Network topology , a term used to describe configurations of computer or biological networks. Topology musical ensemble , an Australian post classical quintet Geospatial topology is the study or science of places with applications in earth science , geography , human geography , and geomorphology . In geographic information system s and their data structures, the terms Geospatial topologytopology and planar enforcement are used to indicate that the border line between two neighboring areas and the border point between two connecting lines is stored only once. Thus, any rounding errors might move the border, but will not lead to gaps or overlaps between the areas. Also in cartography, a topological map is a much simplified map that preserves the mathematical topology while sacrificing scale and shape Topology is often confused with the geographic meaning of topography originally the study of places . The confusion may be a factor in topographies having become confused with terrain or relief , such that they are essentially synonymous. In phylogenetics , the branching pattern of a phylogenetic tree. TopologiLinux , a Linux distribution disambig bar Topologie de Topologie es Topolog a desambiguaci n ... more details
The topology of an electronic circuit is the form taken by the Network analysis electrical circuits network ... are regarded as being the same topology. Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely ... and low pass topologies even though the network topology is identical. A more correct term for these classes ... value is Prototype filter prototype network . Mathematical topology Electronic network topology is related to topology mathematical topology , in particular, for networks which contain only two terminal devices, circuit topology can be viewed as an application of graph theory . In a Network analysis ... are the edge graph theory edges of graph theory. Two networks of this kind have the same topology ... branches in both circuits. Topology names Many topology names relate to their appearance ... topology equivalents.svg thumb 700px center All these topologies are identical. Series topology is a general ... is a common name for the topology in filter design. For a network with three branches there are four ... with three branches Note that the parallel series topology is another representation of the Delta topology discussed below. Series and parallel topologies can continue to be constructed with greater ... rules. The Y topology is also called star topology. However, star topology may also refer to the more ... main Electronic filter topology Image Filter topologies.svg 425px left The topologies shown opposite ... section is identical topology to the potential divider topology. The T section is identical topology to the Y topology. The section is identical topology to the topology. All these topologies can be viewed as a short section of a ladder topology . Longer sections would normally be described as ladder topology. These kinds of circuits are commonly analysed and characterised in terms of a two port network . clear Bridge topology Main Bridge circuit Image Depictions of bridge topology.svg 750px ... 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
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
In mathematics and theoretical computer science the Lawson topology , named after J. D. Lawson, is a topology on partially ordered set s used in the study of domain theory . The lower topology on a poset P is generated by the subbasis consisting of all complements of principal filter mathematics filters on P . The Lawson topology on P is the smallest common refinement of the lower topology and the Scott topology on P . Properties If P is a complete upper semilattice , the Lawson topology on P is always a complete T sub 1 sub topology. See also Scott continuity References G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott 2003 , Continuous Lattices and Domains , Encyclopedia of Mathematics and its Applications, Cambridge University Press. ISBN 0 521 80338 1 External links http www.entcs.org files mfps19 83011.pdf How Do Domains Model Topologies? , Pawel Waszkiewicz, Electronic Notes in Theoretical Computer Science 83 2004 topology stub Category Domain theory Category General topology ... more details
Unreferenced stub auto yes date December 2009 Orphan date December 2009 A topology table is used by router computing router s that route traffic in a network. It consists of all routing tables inside the Autonomous system Internet Autonomous System where the router is positioned. Each router using the routing protocol EIGRP then maintains a topology table for each configured network protocol all routes learned, that are leading to a destination are found in the topology table. EIGRP must have a reliable connection. DEFAULTSORT Topology Table Category Routing Category Network topology Table Compu network stub ... more details
In topology and related areas of mathematics , an induced topology on a topological space is a topology which is optimal for some Function mathematics function from to this topological space. Definition Let math X 0, X 1 math be sets, math f X 0 to X 1 math . If math tau 0 math is a topology on math X 0 math , then a topology induced on math X 1 math by math f math is math U 1 subseteq X 1 f 1 U 1 in tau 0 math . If math tau 1 math is a topology on math X 1 math , then a topology induced on math X 0 math by math f math is math f 1 U 1 U 1 in tau 1 math . The easy way to remember the definitions above is to notice that finding an inverse image is used in both. This is because inverse image preserves union set theory union and intersection set theory intersection . Finding a direct image does not preserve intersection in general. Here is an example where this becomes a hurdle. Consider a set math X 0 2, 1, 1, 2 math with a topology math 2, 1 , 1, 2 math , a set math X 1 1, 0, 1 math and a function math f X 0 to X 1 math such that math f 2 1, f 1 0, f 1 0, f 2 1 math . A set of subsets math tau 1 f U 0 U 0 in tau 0 math is not a topology, because math 1, 0 , 0, 1 subseteq tau 1 math but math 1, 0 cap 0, 1 notin tau 1 math . Properties A topology math tau 1 math induced on math X 1 math by math f math is the finest topology such that math f math is Continuity topology continuous math X 0, tau 0 to X 1, tau 1 math . A topology math tau 0 math induced on math X 0 math by math f math is the coarsest topology such that math f math is continuous math X 0, tau 0 to X 1, tau 1 math . Examples In particular, if math f math is an inclusion map , then math tau 0 math is a subspace topology . References cite book last1 Hu first1 Sze Tsen authorlink1 last2 first2 authorlink2 title Elements of general topology url edition series volume year 1969 publisher Holden Day location isbn id Category Topology Category General topologytopology stub ... more details
In mathematics , the poset topology associated with a partially ordered set S or poset for short is the Alexandrov topology open sets are upper set s on the poset of finite chains of S, ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex is a set of finite sets of vertices, known as faces math sigma subseteq V math , such that math forall rho, sigma. rho subseteq sigma in Delta Rightarrow rho in Delta math Given a simplicial complex as above, we define a point set topology on by letting a subset math Gamma subseteq Delta math be closed if and only if is a simplicial complex math forall rho, sigma. rho subseteq sigma in Gamma Rightarrow rho in Gamma math This is the Alexandrov topology on the poset of faces of . The order complex associated with a poset, S, has the underlying set of S as vertices, and the finite chains i.e. finite totally ordered subsets of S as faces. The poset topology associated with a poset S is the Alexandrov topology on the order complex associated with S. See also Topological combinatorics External links http arxiv.org abs math 0602226 Poset Topology Tools and Applications Michelle L. Wachs, lecture notes IAS Park City Graduate Summer School in Geometric Combinatorics July 2004 Category General topology Category Order theory topology stub ... more details
for the mathematical journal Geometry & Topology In mathematics , geometry and topology is an umbrella term for geometry and topology , as the line between these two is often blurred, most visibly in Riemannian ... like the Gauss Bonnet theorem and Chern Weil theory . Sharp distinctions between geometry and topology can be drawn, however, as discussed below. It is also the title of a journal Geometry & Topology that covers these topics. Scope It is distinct from geometric topology , which more narrowly involves applications of topology to geometry. It includes Differential geometry and topology Geometric topology including low dimensional topology and surgery theory It does not include such parts of algebraic topology as homotopy theory , but some areas of geometry and topology such as surgery theory, particularly algebraic surgery theory are heavily algebraic. Distinction between geometry and topology Pithily, geometry has local structure or infinitesimal , while topology only has global structure. Alternatively, geometry has continuous moduli, while topology has discrete moduli. By examples, an example of geometry is Riemannian geometry , while an example of topology is homotopy theory . The study of metric space s is geometry, the study of topological space s is topology. The terms are not used ... beyond dimension . So differentiable structures on a manifold is an example of topology. By contrast ... structure is topology. If have non trivial deformations, the structure is said to be flexible , and its ... so studying maps up to homotopy is topology. Similarly, differentiable structures on a manifold is usually a discrete space, and hence an example of topology, but exotic R4 exotic R sup 4 sup s have ... symplectic topology and symplectic geometry . By Darboux s theorem , a symplectic manifold has no local structure, which suggests that their study be called topology. By contrast, the space of symplectic ... Geometry And Topology Category Topology Category Geometry ... more details
In mathematics , the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton mathematics singleton math a math is the order section math a x leq a math for each math a in X math . If math leq math is a partial order, the upper topology is the least order consistent topology in which the open set s are the up set s. The lower topology induced by the preorder is defined similarly in terms of the down set s. The preoder inducing the upper topology is its specialization preorder , but the specialization preorder of the lower topology is opposite to the inducing preorder. The real upper topology is most naturally defined on the upper extended real line math infty, infty mathbb R cup infty math by the system math a, infty a in mathbb R cup pm infty math of open sets. Similarly, the real lower topology math infty,a a in mathbb R cup pm infty math is naturally defined on the lower real line math infty, infty mathbb R cup infty math . A real function on a topological space is upper semi continuous if and only if it is lower continuous, i.e. is Continuous function continuous with respect to the lower topology on the lower extended line math infty, infty math . Similarly, a function into the upper real line is lower semi continuous if and only if it is upper continuous, i.e. is Continuous function continuous with respect to the upper topology on math infty, infty math . References cite book author Gerhard Gierz coauthors K.H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott title Continuous Lattices and Domains publisher Cambridge University Press date 2003 isbn 0 521 80338 1 page 510 cite book last Kelley first John L. authorlink John L. Kelley title General Topology publisher Van Nostrand Reinhold date 1955 page 101 cite book last Knapp first Anthony W. title Basic Real Analysis publisher Birkhhauser date 2005 isbn 0817632506 page 481 Category General topology Category Order theory topology stub ... more details
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 ... topology. The weak and strong topologies Let K be a topological field , namely a field mathematics field with a topological space topology such that addition, multiplication, and division are continuity topology continuous . In most applications K will be either the field of complex numbers or the field ..., X 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 ... 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 open ... 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 &phi ... in the weak topology if and only if it can be written as a union of possibly infinitely many sets ..., 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 continuous dual space of X , then the weak topology with respect to F coincides with the weak topology defined above. If the field K has an absolute value math cdot math , then the weak topology &sigma ... more details
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