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
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
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
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
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
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
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
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
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
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
infobox journal title Journal of Topology cover image Journal of Topology cover.gif editor discipline Topology abbreviation J. Topology publisher Oxford University Press on behalf of the London Mathematical Society impact 0.885 impact year 2010 frequency Quarterly history 2008&mdash present website http jtopol.oxfordjournals.org link1 http jtopol.oxfordjournals.org content current link1 name Online access link2 http jtopol.oxfordjournals.org archive link2 name Online archive ISSN 1753 8424 LCCN 2008210020 OCLC 643146824 The Journal of Topology is a Peer review peer reviewed scientific journal on the subject of topology . It was established in 2008, when the editorial board of Topology journal Topology resigned to start the Journal of Topology . ref cite journal url http www.ams.org notices 200705 comm toped web.pdf last Jackson first Allyn title Jumping Ship Topology Board Resigns journal Notices of the American Mathematical Society year 2007 month May pages 637 639 accessdate 26 February 2012 ref The journal is published by Oxford University Press on behalf of the London Mathematical Society . Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews , Science Citation Index , and Zentralblatt MATH . References reflist External link official http jtopol.oxfordjournals.org http www.lms.ac.uk content jtop Journal page at society website DEFAULTSORT topology Category Mathematics journals Category Publications established in 2008 Category Quarterly journals Category Oxford University Press academic journals Category English language journals ... more details
In commutative algebra , the constructible topology on the spectrum of a ring spectrum math operatorname Spec A math of a commutative ring math A math is a topology where each closed set is the image of math operatorname Spec B math in math operatorname Spec A math for some Algebra ring theory algebra B over A . An important feature of this construction is that the map math operatorname Spec B to operatorname Spec A math is a closed map with respect to the constructible topology. With respect to this topology, math operatorname Spec A math is a compact set Definition compact ref Some authors prefer the term quasicompact here. ref , Hausdorff , and totally disconnected topological space . In general the constructible topology is a finer topology than the Zariski topology , but the two topologies will coincide if and only if math A operatorname nil A math is a von Neumann regular ring , where math operatorname nil A , math is the nilradical of a ring nilradical of A . See also Constructible set topology References Reflist Citation last1 Atiyah first1 Michael Francis author1 link Michael Atiyah last2 Macdonald first2 I.G. author2 link Ian G. Macdonald title Introduction to Commutative Algebra publisher Westview Press isbn 978 0 201 40751 8 year 1969 page 50 Citation last Knight first J. T. authorlink title Commutative Algebra publisher Cambridge University Press isbn 0 521 108193 9 year 1971 pages 121 123 topology stub Category Commutative algebra Category Topology ... more details
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 ... X is continuous function topology continuous . X is path connected and so connected space connected ... 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 general topology and related areas of mathematics , the final topology or strong topology or colimit topology or inductive topology on a Set mathematics set math X math , with respect to a family of functions into math X math , is the finest topology on X which makes those functions continuous function topology continuous . Definition Given a set math X math and a family of topological space s math Y i math with functions math f i Y i to X math the final topology math tau math on math X math is the finest topology such that each math f i Y i to X, tau math is continuous function topology continuous . Explicitly, the final topology may be described as follows a subset U of X is open if and only if math f i 1 U math is open in Y sub i sub for each i &isin I . Examples The quotient topology is the final topology on the quotient space with respect to the quotient map . The disjoint union topology disjoint union is the final topology with respect to the family of canonical injection s. More generally, a topological space is coherent topology coherent with a family of subspaces if it has the final topology coinduced by the inclusion maps. The direct limit of any direct system mathematics ... topology determined by the canonical morphisms. Given a family of sets family of topologies &tau sub i sub on a fixed set X the final topology on X with respect to the functions id sub X sub X , &tau ... on X . That is, the final topology &tau is the intersection set theory intersection of the topologies ... under f sub i sub is closed open in math Y i math for each i &isin I . The final topology on X ... I . Image FinalTopology 01.png center Characteristic property of the final topology By the universal property of the disjoint union topology we know that given any family of continuous maps f sub ... f will be a quotient map if and only if X has the final topology determined by the maps f sub i sub . Categorical description In the language of category theory , the final topology construction ... more details
Unreferenced date December 2009 In the mathematics mathematical field of topology , the signature is an integer Invariant mathematics invariant which is defined for an oriented manifold M of dimension d 4 k divisible by four doubly even dimensional . This invariant of a manifold has been studied in detail, starting with Rokhlin s theorem for 4 manifolds. Definition Given a connected space connected and orientable manifold M of dimension 4 k , the cup product gives rise to a quadratic form Q on the middle real cohomology group H sup 2 k sup M , Z . The basic identity for the cup product math alpha p smile beta q 1 pq beta q smile alpha p math shows that with p q 2 k the product is symmetric . It takes values in H sup 4 k sup M , Z . If we assume also that M is compact manifold compact , Poincar duality identifies this with H sub 0 sub M , Z , which can be identified with Z . Therefore cup product, under these hypotheses, does give rise to a symmetric bilinear form on H sup 2 k sup M , Z and therefore to a quadratic form Q . The form Q is non degenerate due to Poincar duality, as it pairs non degenerately with itself. More generally, the signature can be defined in this way for any general compact polyhedron with 4n dimensional Poincar duality. The signature of M is by definition the signature quadratic form signature of Q . If M is not connected, its signature is defined to be the sum of the signatures of its connected components. Other dimensions details L theory If M has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in L theory the signature can be interpreted as the 4 k dimensional simply connected symmetric ... is a cobordism invariant, and in particular is given by some linear combination of its Lev Semenovich Pontryagin Pontryagin numbers. Friedrich Hirzebruch 1954 found an explicit expression for this linear ... Signature Topology Category Geometric topology Category Quadratic forms ... 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
Topology control is a technique used mainly in wireless ad hoc and sensor networks to reduce the initial network topologytopology of the network in order to save energy, cut down interference and extend ..., preserving the saved resources for future maintenance. Topology construction and maintenance Lately, topology control have been divided into two subproblems topology construction , in charge of the initial reduction, and topology maintenance , in charge of the maintenance of the reduced topology so characteristics like connectivity and coverage are preserved. This is the first stage of a topology control protocol. Once the initial topology is deployed, specially when the location of the nodes .... By modifying this parameters, the topology of the network can change. Upon the same time a topology ... The optimal reduced topology stops being it at the first second of full activity. After some time being .... Topology construction algorithms There are many ways to perform topology construction Change ... Clustering, etc. Some examples of topology construction algorithms are Tx range based Geometry based ... Direction Based Yao graph and Nearest neighbor graph , Cone Based Topology Control CBTC , Distributed ... xpl freeabs all.jsp?arnumber 4697849 , A3 A topology construction protocol for WSN ref , EECDS ref ... topology File Capture mst.PNG Reduced network topology via Minimal Spanning Tree Change in Tx Range File Capture reduce.PNG Reduced network topology via Connected Dominating Set Select a subset of nodes that cover all the network and turn off non selected nodes gallery Topology maintenance algorithms In the same manner as topology construction, there are many ways to perform topology maintenance ... examples of topology maintenance algorithms are br Global DGTRec Dynamic Global Topology Recreation Periodically, wake up all inactive nodes, reset the existing reduced topology in the network and apply a topology construction protocol. br SGTRot Static Global Topology Rotation Initially, the topology ... 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. R , T is Hausdorff space Hausdorff but not Regular space regular . The fact that it is Hausdorff follows from the first property. It is not regular since the closed set K and the point 0 have no disjoint Neighbourhood neighbourhoods about them 4. Surprisingly enough, R , T is a Connected space connected ... 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
DISPLAYTITLE h topology In algebraic geometry , the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology theory homology of scheme mathematics scheme . It has several variants, such as the qfh and cdh topologies. Definition Define a morphism of schemes to be submersive or a topological epimorphism if it is surjective on points and its codomain has the quotient topology , i.e., a subset of the codomain is open if and only if its preimage is open. A morphism is universally submersive or a universal topological epimorphism if it remains a topological epimorphism after any base change. ref SGA I, Expos IX, d finition 2.1 ref ref Suslin and Voevodsky, 4.1 ref The covering morphisms of the h topology are the universal topological epimorphisms. The qfh topology has the further restriction that its covering morphisms must be quasi finite. The proper cdh topology is defined as follows. Let nowrap p Y &rarr X be a proper morphism. Suppose that there exists a closed immersion nowrap e A &rarr X . If the morphism nowrap p sup &minus 1 sup X &minus e A &rarr X &minus e A is an isomorphism, then p is a covering morphism for the cdh topology. The cd stands for completely decomposed in the same sense it is used for the Nisnevich topology . An equivalent definition of a covering morphism is that it is a proper morphism p such that for any point x of the codomain, the fiber p sup &minus 1 sup x contains a point rational over the residue field of x . The cdh topology is the smallest Grothendieck topology whose covering morphisms include those of the proper cdh topology and those of the Nisnevich topology. Notes reflist References Andrei Suslin Suslin, A. , and Voevodsky, V., Relative cycles and Chow sheaves , April 1994, http www.math.uiuc.edu K theory 35 . Category Algebraic geometry ... 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
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 , the cartesian product of topological space s can be given several different topologies. The canonical one is the product topology , because it fits rather nicely with the Category theory categorical notion of a product category theory product . Another possibility is the box topology . The box topology has a somewhat more obvious definition than the product topology, but it satisfies fewer desirable properties. In general, the box topology is finer topology finer than the product topology ... many of the factors are trivial topology trivial . Definition Given X such that math X prod ... , index set indexed by math i in I math , the box topology on X is generated by B U sub i sub U ... like boxes or unions thereof. It is easily verified that B is actually a basis topology basis for the topology. Properties Box topology on R sup sup The box topology is completely regular The box topology is neither compact space compact nor Connection mathematics connected The box topology is not first countable Neither is the box topology separable space separable The box topology is paracompact ... of functions from S to X the product topology yields the topology of pointwise convergence ... topology, once again due to its great profusion of open sets, makes convergence very hard. One way to visualize the convergence in this topology is to think of functions from R to R &mdash a sequence of functions converges to a function f in the box topology if, when looking at the graph ... in the product topology as well here we only require all the functions to jump through any given finite set of hoops. This stems directly from the fact that, in the product topology, almost all ... of pointwise convergence. Comparison with product topology The basis sets in the product topology ... U sub i sub are equal to the whole space X sub i sub . The product topology satisfies a very desirable ... by the component functions f sub i sub is continuous function topology continuous if and only ... 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, and especially general topology , the Euclidean topology is an example of a topology given to the set of real number s, denoted by R . To give the set R a topology means to say which subset s of R are open , and to do so in a way that the following axiom s are met ref name CEIT Citation first L. A. last Steen first2 J. A. last2 Seebach title Counterexamples in Topology publisher Dover year 1995 ISBN 048668735X ref The union mathematics union of open sets is an open set. The finite intersection mathematics intersection of open sets is an open set. The set R and the empty set are open sets. Construction The set R and the empty set are required to be open sets, and so we define R and to be open sets in this topology. Given two real numbers, say x and y , with nowrap 1 x y we define an uncountably infinite family of open sets denoted by S sub x , y sub as follows ref name CEIT math S x,y r in bold R x r y . math Along with the set R and the empty set , the sets S sub x , y sub with nowrap 1 x y are used as a basis topology basis for the Euclidean topology. In other words, the open sets of the Euclidean topology are given by the set R , the empty set and the unions and finite intersections of various sets S sub x , y sub for different pairs of x , y . Properties The real line, with this topology, is a T5 space T sub 5 sub space . Given two subsets, say A and B , of R with nowrap 1 font style text decoration overline A font B A font style text decoration overline B font , where font style text decoration overline A font denotes the closure topology closure of A , etc., there exist open sets S sub A sub and S sub B sub with nowrap 1 A S sub A sub and nowrap 1 B S sub B sub such that nowrap 1 S sub A sub S sub B sub . ref name CEIT References Reflist Category Topology es Topolog a euclideana nl Euclidische topologie ... more details
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