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- Directed sets are very useful in topology. We will explore a couple of their uses in this lesson. ...opology, since it is an ordinal, and <math>\{0,1\}</math> has the discrete topology. Then <math>f</math> is not continuous but every sequence in <math>[0,\ome ...6 KB (1,062 words) - 11:53, 21 December 2022
- ==Algebraic topology== ...roximation theorems for topological spaces. A central problem in algebraic topology is to find algebraic invariants of topological spaces, which is usually car ...11 KB (1,301 words) - 12:39, 6 May 2023
- consists of a set <math>X</math> and a `topology' on <math>X</math> where the latter topology in terms of open or of closed sets, a compact-open ...4 KB (550 words) - 16:18, 12 September 2020
- ==What is a Topology?== ...ir intersections, unions and complements. In the basic sense, [[w:Topology|Topology]] (the subject) is the study of structures arising from or related to topol ...9 KB (1,544 words) - 23:36, 4 March 2025
- ...f the subdiscipline [https://de.wikipedia.org/wiki/Topologie_(mathematics) topology] of [https://de.wikipedia.org/wiki/Mathematik mathematics]. By introducing to many and very general structures (such as the topology of function spaces). ...19 KB (3,218 words) - 12:42, 25 November 2024
- ...n [[Introduction to Topology/Lesson 1|the first lesson]] we learned what a topology is, what open sets, closed sets, and bases are. You should be comfortable #If <math>X</math> is any space and <math>Y</math> has the indiscrete topology, then any function <math>f:X\to Y</math> is continuous. ...6 KB (1,037 words) - 05:19, 12 January 2016
- ...ace]] has very many useful properties and they are very good for beginning topology students to study because of their relative intuitive nature. ...collection <math>\{B(x,r)\mid x\in X,\ r>0\}</math> as a basis. Call this topology <math>\mathcal T_d.</math> ...3 KB (568 words) - 05:31, 24 June 2018
- ...ts. For example, in any space with the [[w:indiscrete topology|indiscrete topology]], every sequence converges to every point of the space! ...ty</math> (called the [[w:Cofiniteness#Cofinite_topology|finite complement topology]]). Let <math>x_n=n</math>. Then we see that <math>x_n\to x</math> for an ...3 KB (516 words) - 11:54, 21 December 2022
- ''A variable network topology'' is defined by an indexed family ...462 bytes (67 words) - 16:18, 12 September 2020
- ===Category theory concepts in algebraic topology=== .../|non-Abelian categories]] and [[../ModuleAlgebraic/|non-Abelian algebraic topology]] #supplemental axioms for an Abelian category ...7 KB (684 words) - 07:29, 31 October 2023
- ==An Overview of Algebraic Topology topics== ...roximation theorems for topological spaces. A central problem in algebraic topology is to find algebraic invariants of topological spaces, which is usually car ...3 KB (274 words) - 12:39, 6 May 2023
- ==Algebraic topology topics== ''[[../ModuleAlgebraic/|algebraic topology]]'' (AT) utilizes [[../CoIntersections/|algebraic]] approaches to solve [[. ...10 KB (1,003 words) - 12:39, 6 May 2023
- ...]] and [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]]. ..../NAQAT2/|nonabelian algebraic topology]] (NAAT)'s generalized approach to topology in terms of [[../GroupoidHomomorphism2/|groupoids]] and [[../CubicalHigherH ...6 KB (711 words) - 12:39, 6 May 2023
- ==Non--Abelian algebraic topology== ...tersections/|algebraic]] treatments of topological invariants in Algebraic Topology. ...3 KB (347 words) - 15:46, 12 September 2020
- *[[../CubicalHigherHomotopyGroupoid/|algebraic topology]], *[[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]], ...65 KB (9,661 words) - 04:57, 12 September 2020
- ==Non-Abelian Quantum Algebraic Topology (NAQAT)== ...ned with the applications of [[../CubicalHigherHomotopyGroupoid/|algebraic topology]] methods, results and constructions (including its extensions to [[../Triv ...4 KB (564 words) - 15:46, 12 September 2020
- *[[../CubicalHigherHomotopyGroupoid/|algebraic topology]], *[[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]], ...66 KB (9,854 words) - 12:46, 6 May 2023
- Algebraic topology, quantum algebraic topology, n-logic algebraic categories, ...ant to [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]]. A more extensive bibliography on ''[[../TrivialGroupoid/|category theory ...42 KB (6,254 words) - 04:48, 12 September 2020
- ==topic on the Algebraic Foundations of Quantum Algebraic Topology== ...ons of [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]] ([[../QuantumOperatorAlgebra5/|QAT]]) ...10 KB (1,384 words) - 16:14, 12 September 2020
- *[[../ModuleAlgebraic/|algebraic topology]], *[[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]], ...66 KB (9,841 words) - 12:46, 6 May 2023
Page text matches
- ==An Overview of Algebraic Topology topics== ...roximation theorems for topological spaces. A central problem in algebraic topology is to find algebraic invariants of topological spaces, which is usually car ...3 KB (274 words) - 12:39, 6 May 2023
- ...u</math> then also the finite intersetion of these sets are element of the topology, i.e. ...) then also the union of these sets <math>U_i</math> is an element of the topology <\math>, i.e. ...1 KB (218 words) - 11:43, 31 December 2017
- ==What is a Topology?== ...ir intersections, unions and complements. In the basic sense, [[w:Topology|Topology]] (the subject) is the study of structures arising from or related to topol ...9 KB (1,544 words) - 23:36, 4 March 2025
- ==Non--Abelian algebraic topology== ...tersections/|algebraic]] treatments of topological invariants in Algebraic Topology. ...3 KB (347 words) - 15:46, 12 September 2020
- consists of a set <math>X</math> and a `topology' on <math>X</math> where the latter topology in terms of open or of closed sets, a compact-open ...4 KB (550 words) - 16:18, 12 September 2020
- Hatcher, A. 2001. [http://www.math.cornell.edu/~hatcher/AT/AT.pdf ''Algebraic Topology'' (textbook on line).], Cambridge University Press; Cambridge, UK., 405 pa Ronald Brown: Topology and Groupoids, BookSurge LLC (2006). ...2 KB (293 words) - 16:14, 12 September 2020
- ...h>\mathbb{H}</math>; of course, one considers the inductive limit (strong) topology to be defined on <math>C_c(\grp)</math>, and then also an [[../QuantumSpinNetworkFunctor2/|operator]] ''weak'' topology to be defined on <math>B(\mathbb{H})</math>. ...1 KB (154 words) - 16:01, 12 September 2020
- ...pplicationsInTheoreticalPhysics/|Bibliography for categories and algebraic topology]] ...pplicationsInTheoreticalPhysics/|Bibliography for categories and algebraic topology]] ...1 KB (163 words) - 04:51, 12 September 2020
- ...ace]] has very many useful properties and they are very good for beginning topology students to study because of their relative intuitive nature. ...collection <math>\{B(x,r)\mid x\in X,\ r>0\}</math> as a basis. Call this topology <math>\mathcal T_d.</math> ...3 KB (568 words) - 05:31, 24 June 2018
- ...ts. For example, in any space with the [[w:indiscrete topology|indiscrete topology]], every sequence converges to every point of the space! ...ty</math> (called the [[w:Cofiniteness#Cofinite_topology|finite complement topology]]). Let <math>x_n=n</math>. Then we see that <math>x_n\to x</math> for an ...3 KB (516 words) - 11:54, 21 December 2022
- with a topology on <math>G</math> such that <math>(x,y) \mapsto xy^{-1}</math> is continuou ...mes G</math> is regarded as a [[topological space]] defined by the product topology. ...1 KB (174 words) - 16:15, 12 September 2020
- ...P</math> are metrizable (topological) [[../TrivialGroupoid/|groups]] whose topology is Polish, and thus they admit a compatible [[../MetricTensor/|metric]] <ma ...are Borel isomorphic to <math>\mathbb{R </math> equipped with the standard topology;} This also implies that all uncountable Polish space have the cardinality ...2 KB (275 words) - 04:57, 12 September 2020
- ...topology, and especially in [[../CubicalHigherHomotopyGroupoid/|algebraic topology]]; they may be considered as one of the simplest, special [[../Bijective/|t ...'' -- concepts in algebraic topology (<ref name='BR2006'>R. Brown. 2006. ''Topology and Groupoids'' . Booksurge PLC.</ref>) ...5 KB (656 words) - 23:58, 30 August 2020
- ...pplicationsInTheoreticalPhysics/|Bibliography for categories and algebraic topology]] ...pplicationsInTheoreticalPhysics/|Bibliography for categories and algebraic topology]] ...1 KB (184 words) - 21:39, 6 March 2022
- ...m a CW-complex associated with the black hole. The <math>CW</math>-complex topology is consistent with both the <math>\mathbf{G}_{Lg}</math>symmetry of the qua ...sections of both physics and [[../CubicalHigherHomotopyGroupoid/|algebraic topology]]. ...2 KB (321 words) - 11:42, 16 June 2023
- ...pplicationsInTheoreticalPhysics/|Bibliography for categories and algebraic topology]] ...pplicationsInTheoreticalPhysics/|Bibliography for categories and algebraic topology]] ...1 KB (206 words) - 04:50, 12 September 2020
- Hausdorff topology, that is, if given any distinct points <math>x,y\in X_{LC}</math>, there ex ...]] in: [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]] ([[../QuantumOperatorAlgebra5/|QAT]]), topological QFT ([[../SUSY2/|TQFT] ...2 KB (290 words) - 04:40, 1 July 2015
- ...]] and [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]]. ...gy]] ([[../NonabelianAlgebraicTopology3/|NAAT]])'s generalized approach to topology in terms of [[../GroupoidHomomorphism2/|groupoids]] and [[../CubicalHigherH ...3 KB (383 words) - 15:47, 12 September 2020
- ==Algebraic topology topics== ''[[../ModuleAlgebraic/|algebraic topology]]'' (AT) utilizes [[../CoIntersections/|algebraic]] approaches to solve [[. ...10 KB (1,003 words) - 12:39, 6 May 2023
- ''A variable network topology'' is defined by an indexed family ...462 bytes (67 words) - 16:18, 12 September 2020