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 mathematics , a strong topology is a topology which is stronger than some other default topology. This term is used to describe different topologies depending on context, and it may refer to the final topology on the disjoint union topology disjoint union the topology arising from a normed vector space norm the strong operator topology the strong topology polar topology , which subsumes all topologies above. Note that a topology is stronger than a topology is a Comparison of topologies finer topology if contains all the open sets of . In algebraic geometry , it usually means the topology of an algebraic variety as complex manifold or subspace of complex projective space , as opposed to the Zariski topology which is rarely even a Hausdorff space . See also Weak topology mathdab Category Topology ... more details
In functional analysis and related areas of mathematics the strong topology is the finer topology finest polar topology , the topology with the most open set s, on a dual pair . The coarser topology coarsest polar topology is called weak topology polar topology weak topology . Definition Given a dual pair math X,Y, langle , rangle math the strong topology math beta Y, X math on math Y math is the polar topology defined by using the family of all sets in math X math where the polar set in math Y math is Absorption law absorbent . Examples Given a normed vector space math X math and its continuous dual math X math then math beta X , X math topology on math X math is identical to the topology induced by the operator norm . Conversely math beta X, X math topology on math X math is identical to the topology induced by the norm mathematics norm . Properties In barrelled space s the strong topology is identical to the Mackey topology . mathanalysis stub Category Topology of function spaces ... more details
In mathematics , the uniform topology on a space has several different meanings depending on the context In functional analysis, it sometimes refers to a polar topology on a topological vector space. In general topology, it is the topology carried by a uniform space . In real analysis, it is the topology of uniform convergence . Disambig ... more details
Unreferenced date December 2009 In functional analysis , a branch of mathematics , the ultraweak topology , also called the weak topology , or weak operator topology or weak topology , on the set B H of bounded operator s on a Hilbert space is the weak topology weak topology obtained from the predual B sub sub H of B H , the trace class operators on H . In other words it is the weakest topology such that all elements of the predual are continuous when considered as functions on B H . Relation with the weak operator topology The ultraweak topology is similar to the weak operator topology. For example, on any norm bounded set the weak operator and ultraweak topologies are the same, and in particular the unit ball is compact in both topologies. The ultraweak topology is stronger than the weak operator topology. One problem with the weak operator topology is that the dual of B H with the weak operator topology is too small . The ultraweak topology fixes this problem the dual is the full predual B sub sub H of all trace class operators. In general the ultraweak topology is more useful than the weak operator topology, but it is more complicated to define, and the weak operator topology is often more apparently convenient. The ultraweak topology can be obtained from the weak operator topology as follows. If H sub 1 sub is a separable infinite dimensional Hilbert space then B H can be embedded in B H H sub 1 sub by tensoring with the identity map on H sub 1 sub . Then the restriction of the weak operator topology on B H H sub 1 sub is the ultraweak topology of B H . See also Topologies on the set of operators on a Hilbert space ultrastrong topology weak operator topology DEFAULTSORT Ultraweak Topology Category Topology of function spaces Category Von Neumann algebras ... more details
Unreferenced date December 2009 In functional analysis and related areas of mathematics the weak topology is the coarser topology coarsest polar topology , the topology with the fewest open set s, on a dual pair . The finer topology finest polar topology is called strong topology polar topology strong topology . Under the weak topology the Bounded set topological vector space bounded set s coincide with the relatively compact set s which leads to the important Bourbaki Alaoglu theorem . Definition Given a dual pair math X,Y, langle , rangle math the weak topology math sigma X,Y math is the weakest polar topology on math X math so that math X, sigma X,Y simeq Y math . That is the continuous dual of math X, sigma X,Y math is equal to math Y math up to isomorphism . The weak topology is constructed as follows For every math y math in math Y math on math X math we define a semi norm on math X math math p y X to mathbb R math with math p y x vert langle x , y rangle vert qquad x in X math This family of semi norms defines a locally convex topology on math X math . Examples Given a normed vector space math X math and its continuous dual math X math , math sigma X, X math is called the weak topology on math X math and math sigma X , X math the weak star topology weak topology on math X math DEFAULTSORT Weak Topology Polar Topology Category Topology of function spaces ... more details
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
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
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 ... 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 References Some standard books on general topology include Bourbaki cite Topologie G n rale cite ... 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 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
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
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
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 , 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 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 topological ... . 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 publisher Prentice ... more details
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