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Concept in algebraic topology
specifically algebraic topology, a pair ( X , A ) {\displaystyle (X,A)} is shorthand for an inclusion of topological spaces i : A ↪ X {\displaystyle i\colon
Topological_pair
Topics referred to by the same term
ring Pair type, in programming languages and type theory, a product type with two component types Topological pair, an inclusion of topological spaces
Pair
Mathematical concept in algebraic topology
In mathematics, particularly in algebraic topology, a taut pair is a topological pair ( A , B ) {\displaystyle (A,B)} , whose cohomology modules H q (
Tautness_(topology)
Type of quantum computer
processors, the first used a toric code with twist defects as a topological degeneracy (or topological defect) while the second used a different but related protocol
Topological_quantum_computer
Node ordering for directed acyclic graphs
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed
Topological_sorting
Superconductor that depends on atomic structure
decoherence. Topological insulator Majorana fermion Kitaev chain Topological quantum computer Cooper pair Dumé, Isabelle (2023-08-03). "Topological superconductor
Topological_superconductor
Property of topological submanifolds
a smoothness condition that can be imposed on topological submanifolds. In the category of topological manifolds, locally flat submanifolds play a role
Local_flatness
Branch of mathematics
invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows
Topology
State of matter with insulating bulk but conductive boundary
material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator. The topological insulator cannot
Topological_insulator
Mathematical property of a space
and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms
Topological_property
Directed graph with no directed cycles
a topological ordering is acyclic. Conversely, every directed acyclic graph has at least one topological ordering. The existence of a topological ordering
Directed_acyclic_graph
statement can be generalized to simple topological graphs. A topological graph is called "simple" if any pair of its edges share at most one point, which
Topological_graph
Branch of mathematics studying (smooth) functions of manifolds
structures on R4. Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable
Geometric_topology
Three-holed sphere
closed surfaces into pairs of pants are used to construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory
Pair_of_pants_(mathematics)
Topological relational characteristic
topological space in which every pair of distinct points is topologically distinguishable. This is the weakest of the separation axioms. Topological
Topological indistinguishability
Topological_indistinguishability
Analysis of datasets using techniques from topology
In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information
Topological_data_analysis
Topological space that is connected
Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X {\displaystyle X} is a connected
Connected_space
Topological quantum error correcting code
quantum double models. It is also the simplest example of topological order—Z2 topological order (first studied in the context of Z2 spin liquid in 1991)
Surface_code
Topological space in which all singleton sets are closed
for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms. Let X be a topological space and let
T1_space
Topologically stable solution of a partial differential equation
In mathematics and physics, solitons, topological solitons and topological defects are three closely related ideas, all of which signify structures in
Topological_defect
Topological space that locally resembles Euclidean space
structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological space locally homeomorphic to
Manifold
Cohomology theory for topological spaces
Edwin H. Spanier (1948) for all topological spaces, based on a suggestion of Alexander D. Wallace. If X is a topological space and G is an R-module where
Alexander–Spanier_cohomology
Number representing system complexity
In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of
Topological_entropy
Vector space with a notion of nearness
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures
Topological_vector_space
Field theory involving topological effects in physics
mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While
Topological quantum field theory
Topological_quantum_field_theory
Pair of electrons bound together at low temperature, allowing for superconductivity
Superinsulator Topological superconductor Unconventional superconductor Lone pair Electron pair Cooper, Leon N. (1956). "Bound electron pairs in a degenerate
Cooper_pair
Homology for a pair of topological spaces
the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology
Relative_homology
2D surface which extends indefinitely
but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent
Plane_(mathematics)
Concept in math
other categories in geometry and algebra. The category of topological spaces Top has topological spaces as objects and as morphisms the continuous maps between
Homotopy_category
Concept in topology
the underlying topological space of any scheme. Given any topological space one can construct a T0 space by identifying topologically indistinguishable
Kolmogorov_space
Mathematical game on a topological space
In mathematics, a topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects
Topological_game
Prime knot with crossing number 10
the July 8, 1986 New York Times. The Perko pair is one of five knots with 10 crossings where the topological and smooth 4-genus are different; the former
Perko_pair
Type of topological space
is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space,
Hausdorff_space
Mathematical description of mixing substances
definitions for mixing exist, including strong mixing, weak mixing and topological mixing, with the last not requiring a measure to be defined. Some of
Mixing_(mathematics)
Branch of algebraic topology
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas
Topological_K-theory
Tensor product constructions for topological vector spaces
there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces
Topological_tensor_product
Mathematical concept
a topological space, it is a connected space. Thus, manifolds, Lie groups, and graphs are all called connected if they are connected as topological spaces
Connectedness
Duality for locally compact abelian groups
main directions: for commutative topological groups that are not locally compact, and for noncommutative topological groups. The theories in these two
Pontryagin_duality
Theory in theoretical physics
In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists
Topological_string_theory
Structure formed by double-stranded molecules
similar topological constraints. For many years, the origin of residual supercoiling in eukaryotic genomes remained unclear. This topological puzzle was
Nucleic_acid_double_helix
In mathematics, vector space of linear forms
of topological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space". For a topological vector
Dual_space
Generalization of boundedness
Kolmogorov in 1935. Suppose X {\displaystyle X} is a topological vector space (TVS) over a topological field K . {\displaystyle \mathbb {K} .} A subset B
Bounded set (topological vector space)
Bounded_set_(topological_vector_space)
Algebraic structure with "nice" duality properties
interest has been renewed in Frobenius algebras due to connections to topological quantum field theory. A finite-dimensional, unital, associative algebra
Frobenius_algebra
Generalized topological space
continuous function f: X → X' between two topological spaces becomes an adjoint pair (f,g) in which f is now paired with a realization of the continuity condition
Chu_space
Non-orientable surface with one edge
Every non-orientable surface contains a Möbius strip. As an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean
Möbius_strip
Volume space bounded by a sphere
talk about balls in any topological space X, not necessarily induced by a metric. An (open or closed) n-dimensional topological ball of X is any subset
Ball_(mathematics)
Concept in topology
In mathematics, topological complexity of a topological space X (also denoted by TC(X)) is a topological invariant closely connected to the motion planning
Topological_complexity
Algebraic structure associated with a topological space
important in the study of topological spaces. Under nice conditions in which distinct homology theories for a single topological space produce the same homology
Homology_(mathematics)
Type of relation for subsets of a topological space
and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way:
Separated_sets
Algebraic structure in linear algebra
{\displaystyle V\to W,} maps between topological vector spaces are required to be continuous. In particular, the (topological) dual space V ∗ {\displaystyle
Vector_space
Self-interacting genomic region
Dixon JR, Selvaraj S, Yue F, Kim A, Li Y, Shen Y, et al. (April 2012). "Topological domains in mammalian genomes identified by analysis of chromatin interactions"
Topologically associating domain
Topologically_associating_domain
Mathematical set with some added structure
linear and topological structures underlie the linear topological space (in other words, topological vector space) structure. A linear topological space is
Space_(mathematics)
Model for predicting molecular geometry
electrostatic repulsion. The insights of VSEPR theory are derived from topological analysis of the electron density of molecules. Such quantum chemical
VSEPR_theory
Property of topological space
Hausdorff space is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two
Regular_space
Dual pair of vector spaces
In mathematics, a dual system, dual pair or a duality over a field K {\displaystyle \mathbb {K} } is a triple ( X , Y , b ) {\displaystyle (X,Y,b)} consisting
Dual_system
Type of topological order in condensed matter physics
entanglement see topological order, which is not related to the famous EPR paradox). Since short-range entangled states have only trivial topological orders we
Symmetry-protected topological order
Symmetry-protected_topological_order
bounded if its image is a bounded set. Category of topological spaces The category Top has topological spaces as objects and continuous maps as morphisms
Glossary_of_general_topology
Algebraic structure formed from a collection of algebraic structures
{B} \end{bmatrix}}.} A topological vector space (TVS) X , {\displaystyle X,} such as a Banach space, is said to be a topological direct sum of two vector
Direct_sum
Theorem in dynamical systems theory
systems theory, Conley index theory, named after Charles Conley, analyzes topological structure of invariant sets of diffeomorphisms and of smooth flows. It
Conley_index_theory
on 2g-2+n the Euler characteristics, whence the name "topological recursion". The topological recursion was first discovered in random matrices. One
Topological_recursion
Mathematical space with a notion of distance
different metric properties. Conversely, not every topological space can be given a metric. Topological spaces which are compatible with a metric are called
Metric_space
Tool to track locally defined data attached to the open sets of a topological space
{\displaystyle F} arises from a natural topological situation, E {\displaystyle E} may not have any clear topological interpretation. For example, if F {\displaystyle
Sheaf_(mathematics)
Branch of topology
topology. A set with a topology is called a topological space. Metric spaces are an important class of topological spaces where a real, non-negative distance
General_topology
Way of decomposing a topological space
an abstract stratified space, or a Thom–Mather stratified space is a topological space X that has been decomposed into pieces called strata; these strata
Thom–Mather_stratified_space
Topological space that is maximally disconnected
disconnected space is a topological space that has only singletons (and the empty set) as connected subsets. In every topological space, the singletons
Totally_disconnected_space
Topological vector space whose topology can be defined by a metric
{\displaystyle X} into a topological vector space). Every topological vector space (TVS) X {\displaystyle X} is an additive commutative topological group but not
Metrizable topological vector space
Metrizable_topological_vector_space
Infinitely detailed mathematical structure
curve map is not a homeomorphism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert
Fractal
Category whose objects are topological spaces and whose morphisms are continuous maps
the category of topological spaces, often denoted T o p {\displaystyle \mathbf {Top} } , is the category whose objects are topological spaces and whose
Category of topological spaces
Category_of_topological_spaces
Multiple equivalent ways to define a topological space
arise from a topological structure or not. Such questions are greatly clarified by the topological axioms based on convergence. A topological space is a
Axiomatic foundations of topological spaces
Axiomatic_foundations_of_topological_spaces
Topological spaces whose union is a boundary
domains of topological quantum field theories. Roughly speaking, an n {\displaystyle n} -dimensional manifold M {\displaystyle M} is a topological space locally
Cobordism
Mathematical concept
topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert
Weak_topology
Type of topological space
(also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called cells) of different dimensions
CW_complex
algebraic and topological dual spaces, and ⟨ , ⟩ : X × X ′ → R {\displaystyle \langle ,\rangle :X\times X'\to \mathbb {R} } is the dual pair. A topology
Minlos–Sazonov_theorem
Branch of mathematics
from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though
Algebraic_topology
French mathematician (1928–2014)
Dieudonné and Laurent Schwartz. His key contributions include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational
Alexander_Grothendieck
Topological concept in algebraic geometry
group of topological spaces. In algebraic topology, the fundamental group π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} of a pointed topological space (
Étale_fundamental_group
Group of isotopy classes of a topological automorphism group
a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Consider a topological space
Mapping_class_group
Flat-sided three-dimensional shape
notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks (the faces) whose pairwise
Polyhedron
Phenomenon in many-body quantum systems
perturbations. Topological degeneracy can be used to protect qubits which allows topological quantum computation. It is believed that topological degeneracy
Topological_degeneracy
algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known
Fundamental_groupoid
Algebraic theory
algebraic theory of topological quantum information is a collection of algebraic techniques developed and applied to topological aspects of condensed
Algebraic theory of topological quantum information
Algebraic_theory_of_topological_quantum_information
Space where bounded operators are continuous
any bounded subset Bounded set (topological vector space) – Generalization of boundedness Locally convex topological vector space – Space with topology
Bornological_space
Mathematical structure
topological manifold with some additional mathematical structure that allows for differential calculus on the manifold. If M is already a topological
Differential_structure
Topological space formed from distances
also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial
Vietoris–Rips_complex
Generalized sphere of dimension n (mathematics)
-sphere is the topological join of n + 1 {\displaystyle n+1} pairs of isolated points. Intuitively, the topological join of two pairs is generated by
N-sphere
Mathematical group of the homotopy classes of loops in a topological space
the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces
Fundamental_group
Graph drawing used to study Riemann surfaces
graph must be bipartite. The faces of the embedding are required to be topological disks. The surface and the embedding may be described combinatorially
Dessin_d'enfant
Polyhedron associated with another by swapping vertices for faces
kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. In Euclidean space, the dual of a polyhedron P {\displaystyle
Dual_polyhedron
Structure in functional analysis
analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points
Complete topological vector space
Complete_topological_vector_space
Mathematical set with an ordering
topological space, then it is customary to assume that { ( a , b ) : a ≤ b } {\displaystyle \{(a,b):a\leq b\}} is a closed subset of the topological product
Partially_ordered_set
field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper closed subsets
Hyperconnected_space
Properties that homology theories of topological spaces have in common
functors H n {\displaystyle H_{n}} from the category of pairs ( X , A ) {\displaystyle (X,A)} of topological spaces to the category of abelian groups, together
Eilenberg–Steenrod_axioms
Condensed matter phenomenon; vortex-like magnetic quasiparticle
a non-zero, integer value of the topological index, (not to be confused with the chemistry meaning of 'topological index'). This value is sometimes also
Magnetic_skyrmion
Concept in differential geometry
of the topological Euler characteristic that includes contributions coming from nontrivial automorphisms. In particular, unlike a topological Euler characteristic
Euler characteristic of an orbifold
Euler_characteristic_of_an_orbifold
Subset of Euclidean space is compact if and only if it is closed and bounded
to showing that any compact subset S {\displaystyle S} of a Hausdorff topological space X {\displaystyle X} is closed in X {\displaystyle X} . If a set
Heine–Borel_theorem
Mathematical result in differential geometry
dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as
Atiyah–Singer_index_theorem
Theorem in geometry about convex sets
their images under f intersect - as claimed by the new formulation. The topological Radon theorem generalizes this formluation. It allows f to be any continuous
Radon's_theorem
Relationship between two functors abstracting many common constructions
a right adjoint. Let G be the functor from topological spaces to sets that associates to every topological space its underlying set (forgetting the topology
Adjoint_functors
consistent in the overall pattern, like a solid. Topological semimetals: Weyl semimetal Dirac semimetal Topological superconductor Metallic and insulating states
List_of_states_of_matter
Mathematics concept
originally described the topological twisting of the N=(2,2) supersymmetric field theory into what he called the A and B model topological string theories[citation
Homological_mirror_symmetry
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