Category: Topology

Busemann G-space
In mathematics, a Busemann G-space is a type of metric space first described by Herbert Busemann in 1942. If is a metric space such that 1. * for every two distinct there exists such that 2. * every
Bousfield class
In algebraic topology, the Bousfield class of, say, a spectrum X is the set of all (say) spectra Y whose smash product with X is zero: . Two objects are Bousfield equivalent if their Bousfield classes
The toponome is the spatial network code of proteins and other biomolecules in morphologically intact cells and tissues. It is mapped and decoded by imaging cycler microscopy (ICM) in situ able to co-
Bitopological space
In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is and the topologies are and then the bitopological space is referred to as . The notion was introduc
Hopf conjecture
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
Star refinement
In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X. The general definition makes
Tucker's lemma
In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker. Let T be a triangulation of the closed n-dimensional ball . Assume T is antipodally s
Tube lemma
In mathematics, particularly topology, the tube lemma is a useful tool in order to prove that the finite product of compact spaces is compact.
Geometry and topology
In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly,
Subnet (mathematics)
In topology and related areas of mathematics, a subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The de
Delta set
In mathematics, a Δ-set S, often called a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of rela
Thurston norm
In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of hom
Oswald Veblen Prize in Geometry
The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen. The Veblen
Cover (topology)
In mathematics, and more particularly in set theory, a cover (or covering) of a set is a collection of subsets of whose union is all of . More formally, if is an indexed family of subsets , then is a
Prosolvable group
In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently,
Topological ring
In mathematics, a topological ring is a ring that is also a topological space such that both the addition and the multiplication are continuous as maps: where carries the product topology. That means
In the mathematical branch of topology, a hyperspace (or a space equipped with a hypertopology) is a topological space, which consists of the set CL(X) of all closed subsets of another topological spa
Nearly Kähler manifold
In mathematics, a nearly Kähler manifold is an almost Hermitian manifold , with almost complex structure ,such that the (2,1)-tensor is skew-symmetric. So, for every vector field on . In particular, a
In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, su
Ambient space
An ambient space or ambient configuration space is the space surrounding an object. While the ambient space and hodological space are both considered ways of perceiving penetrable space, the former pe
Relative interior
In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces. Fo
Corona set
In mathematics, the corona or corona set of a topological space X is the complement βX\X of the space in its Stone–Čech compactification βX. A topological space is said to be σ-compact if it is the un
Loop (topology)
In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its
Aluthge transform
In mathematics and more precisely in functional analysis, the Aluthge transformation is an operation defined on the set of bounded operators of a Hilbert space. It was introduced by to study linear op
Linear topology
In algebra, a linear topology on a left -module is a topology on that is invariant under translations and admits a fundamental system of neighborhood of that consists of submodules of If there is such
Noncommutative topology
In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts. The term has its origins in the Gelfand–Naimark theorem, which implies the du
Pair of pants (mathematics)
In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a
Inductive tensor product
The finest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) separately continuous is called th
Wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints and ) the wedge
Kline sphere characterization
In mathematics, a Kline sphere characterization, named after John Robert Kline, is a topological characterization of a two-dimensional sphere in terms of what sort of subset separates it. Its proof wa
Abstract cell complex
In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point. The complex is called “abstract”
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of
Matching distance
In mathematics, the matching distance is a metric on the space of size functions. The core of the definition of matching distance is the observation that theinformation contained in a size function ca
T1 process
A T1 process (or topological rearrangement process of the first kind) is one of the main processes by which cellular materials such as foams or biological tissues change shapes. It involves four discr
Compact complement topology
In mathematics, the compact complement topology is a topology defined on the set of real numbers, defined by declaring a subset open if and only if it is either empty or its complement is compact in t
Moduli stack of formal group laws
In algebraic geometry, the moduli stack of formal group laws is a stack classifying formal group laws and isomorphisms between them. It is denoted by . It is a "geometric “object" that underlies the c
Topology (chemistry)
In chemistry, topology provides a way of describing and predicting the molecular structure within the constraints of three-dimensional (3-D) space. Given the determinants of chemical bonding and the c
Leray–Schauder degree
In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres or equivalently to a boundary sphere preserving continuous maps betwee
Menger space
In mathematics, a Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Menger space is a space in which for every sequence of open c
Quantum topology
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology. Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a frame
Milnor's sphere
In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnorpg 14 was trying to understand the structure of -connected manifolds of dimension (since -connected -
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simple
Toric manifold
In mathematics, a toric manifold is a topological analogue of toric variety in algebraic geometry. It is an even-dimensional manifold with an effective smooth action of an -dimensional compact torus w
Cubical set
In topology, a branch of mathematics, a cubical set is a set-valued contravariant functor on the category of (various) n-cubes. See the references for the more precise definitions.
First International Topological Conference
The First International Topological Conference was held in Moscow, 4–10 September, 1935. With presentations by topologists from 10 different countries it constituted the first genuinely international
Akbulut cork
In topology, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut. A compa
Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space r
Quasi-relative interior
In topology, a branch of mathematics, the quasi-relative interior of a subset of a vector space is a refinement of the concept of the interior. Formally, if is a linear space then the quasi-relative i
Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topolo
Retraction (topology)
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then cal
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'globe, ball') is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the
Heat kernel signature
A heat kernel signature (HKS) is a feature descriptor for use in deformable shape analysis and belongs to the group of spectral shape analysis methods. For each point in the shape, HKS defines its fea
Shelling (topology)
In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another simplex) in a well-behaved way. A complex admitti
Double vector bundle
In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent of a vector bundle and the double tangent bundle .
Collectionwise Hausdorff space
In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete subset of , there is a pairwise disjoint family of open sets with each
Size theory
In mathematics, size theory studies the properties of topological spaces endowed with -valued functions, with respect to the change of these functions. More formally, the subject of size theory is the
Periodic table of topological invariants
The periodic table of topological invariants is an application of topology to physics. It indicates the group of topological invariant for topological insulators and superconductors in each dimension
Pushforward (homology)
In algebraic topology, the pushforward of a continuous function : between two topological spaces is a homomorphism between the homology groups for . Homology is a functor which converts a topological
List of topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology o
Extension of a topological group
In mathematics, more specifically in topological groups, an extension of topological groups, or a topological extension, is a short exact sequence where and are topological groups and and are continuo
Equivariant topology
In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps , and while equivariant top
Priestley space
In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play
Verdier duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by Jean-Louis Verdier as an ana
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, al
Exterior space
In mathematics, the notion of externology in a topological space X generalizes the basic properties of the family εXcc = {E ⊆ X : X\E is a closed compact subset of X} of complements of the closed comp
Fixed-point space
In mathematics, a Hausdorff space X is called a fixed-point space if every continuous function has a fixed point. For example, any closed interval [a,b] in is a fixed point space, and it can be proved
Descent along torsors
In mathematics, given a G-torsor X → Y and a stack F, the descent along torsors says there is a canonical equivalence between F(Y), the category of Y-points and F(X)G, the category of G-equivariant X-
Cool S
The Cool S (also known as the Stussy S, Super S, Superman S, Universal S, Pointy S, Middle School S, Graffiti S, and by many other names) is a graffiti sign in popular culture that is typically doodle
Double tangent bundle
In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πT
In mathematics, a topological space is a D-space if for any family of open sets such that for all points , there is a closed discrete subset of the space such that .
Descent (mathematics)
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with so
Antipodal point
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through
Dispersion point
In topology, a dispersion point or explosion point is a point in a topological space the removal of which leaves the space highly disconnected. More specifically, if X is a connected topological space
Clutching construction
In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
Topological Galois theory
In mathematics, topological Galois theory is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by V. I. Arnold and concerns the applications of some
Loop space
In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with
Metric space
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Met
Universal space
In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics.
Accumulation point
In mathematics, a limit point, accumulation point, or cluster point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with res
Order unit
An order unit is an element of an ordered vector space which can be used to bound all elements from above. In this way (as seen in the first below) the order unit generalizes the unit element in the r
N-topological space
In mathematics, an N-topological space is a set equipped with N arbitrary topologies. If τ1, τ2, ..., τN are N topologies defined on a nonempty set X, then the N-topological space is denoted by (X,τ1,
Analysis Situs (book)
Analysis Situs is a book by the Princeton mathematician Oswald Veblen, published in 1922. It is based on his 1916 lectures at the Cambridge Colloquium of the American Mathematical Society. The book, w
Genus g surface
In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g many tori: the interior of a disk is removed from each of g many tori and the
Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations al
In mathematics, a subpaving is a set of nonoverlapping boxes of R⁺. A subset X of Rⁿ can be approximated by two subpavings X⁻ and X⁺ such that X⁻ ⊂ X ⊂ X⁺. In R¹ the boxes are line segments, in R² rec
In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is ro
Shape theory (mathematics)
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape th
Bounding point
In functional analysis, a branch of mathematics, a bounding point of a subset of a vector space is a conceptual extension of the boundary of a set.
Locally normal space
In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that e
Coarse structure
In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structu
Pachner moves
In topology, a branch of mathematics, Pachner moves, named after Udo Pachner, are ways of replacing a triangulation of a piecewise linear manifold by a different triangulation of a homeomorphic manifo
Shape analysis (digital geometry)
This article describes shape analysis to analyze and process geometric shapes.
Suspension (topology)
In topology, a branch of mathematics, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points. One views X as "susp
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on -dimensional Euclidean space by the Euclidean metric.
In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.
Non-orientable wormhole
In wormhole theory, a non-orientable wormhole is a wormhole connection that appears to reverse the chirality of anything passed through it. It is related to the "twisted" connections normally used to
Topological recursion
In mathematics, topological recursion is a recursive definition of invariants of spectral curves.It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory,
Ambient isotopy
In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another s
Excisive triad
In topology, a branch of mathematics, an excisive triad is a triple of topological spaces such that A, B are subspaces of X and X is the union of the interior of A and the interior of B. Note B is not
Profinite word
In mathematics, more precisely in formal language theory, the profinite words are a generalization of the notion of finite words into a complete topological space. This notion allows the use of topolo
Sperner's lemma
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring
Rothberger space
In mathematics, a Rothberger space is a topological space that satisfies a certain a basic selection principle. A Rothberger space is a space in which for every sequence of open covers of the space th
Open book decomposition
In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books
In musical tuning and harmony, the Tonnetz (German for 'tone network') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representation
Pseudonormal space
In mathematics, in the field of topology, a topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing the
Covering number
In mathematics, a covering number is the number of spherical balls of a given size needed to completely cover a given space, with possible overlaps. Two related concepts are the packing number, the nu
Principal geodesic analysis
In geometric data analysis and statistical shape analysis, principal geodesic analysis is a generalization of principal component analysis to a non-Euclidean, non-linear setting of manifolds suitable
In general topology, a subset of a topological space is said to be dense-in-itself or crowdedif has no isolated point.Equivalently, is dense-in-itself if every point of is a limit point of .Thus is de
Linear continuum
In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set S of more than one element that
Eilenberg–Maclane spectrum
In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra for any Abelian group pg 134. Note, this construction can be generalized to
Non-Hausdorff manifold
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (ex
Curve complex
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S. The curve complex turned out to be a
Algebraic interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior.
Constructible topology
In commutative algebra, the constructible topology on the spectrum of a commutative ring is a topology where each closed set is the image of in for some algebra B over A. An important feature of this
Brown–Gitler spectrum
In topology, a discipline within mathematics, the Brown–Gitler spectrum is a spectrum whose cohomology is a certain cyclic module over the Steenrod algebra. Brown–Gitler spectra are defined by the iso
Simplicial sphere
In geometry and combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as the boundaries of convex poly
Topological module
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
Modes of convergence
In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are d
Stone duality
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collecte
Stratified space
In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g.,
Map (graph theory)
In topology and graph theory, a map is a subdivision of a surface such as the Euclidean plane into interior-disjoint regions,formed by embedding a graph onto the surface and forming connected componen
Constructible set (topology)
In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure.They are used particularly in algebraic geometry and related fields. A key resul
Train track (mathematics)
In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions: 1. * The curves meet at a finite set of vertices called switches. 2.
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, In contrast, the usual torus is the Cartesian product of on
Locally discrete collection
In mathematics, particularly topology, collections of subsets are said to be locally discrete if they look like they have precisely one element from a local point of view. The study of locally discret
Specialization (pre)order
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in
Shrinking space
In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the
Set inversion
In mathematics, set inversion is the problem of characterizing the preimage X of a set Y by a function f, i.e., X = f  −1(Y ) = {x ∈ Rn | f(x) ∈ Y }. It can also be viewed as the problem of describing
Condensation point
In mathematics, a condensation point p of a subset S of a topological space is any point p such that every neighborhood of p contains uncountably many points of S. Thus "condensation point" is synonym
Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interva
Real tree
In mathematics, real trees (also called -trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and
Convergence space
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain properties relating elements of X with the fam
Spectral shape analysis
Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami
Free loop
In the mathematical field of topology, a free loop is a variant of the mathematical notion of a loop. Whereas a loop has a distinguished point on it, called a basepoint, a free loop lacks such a disti
In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a rando
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each o
Topological geometry
Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations
Dieudonné plank
In mathematics, the Dieudonné plank is a specific topological space introduced by Dieudonné. It is an example of a metacompact space that is not paracompact.
Fσ set
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed) and σ for somme (French: sum, union). The compleme
Separated sets
In topology 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: roughly speaking, neither overlapping
Reach (mathematics)
Let X be a subset of Rn. Then reach of X is defined as
Complementarity theory
A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a function of two vector variables subject to certain requirements
Chu space
Chu spaces generalize the notion of topological space by dropping the requirements that the set of open sets be closed under union and finite intersection, that the open sets be extensional, and that
Fréchet filter
In mathematics, the Fréchet filter, also called the cofinite filter, on a set is a certain collection of subsets of (that is, it is a particular subset of the power set of ). A subset of belongs to th
Kernel (set theory)
In set theory, the kernel of a function (or equivalence kernel) may be taken to be either * the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as
Locally compact field
In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space. These kinds of fields were originally introduced in p-adic analysis since the fields
Cauchy sequence
In mathematics, a Cauchy sequence (French pronunciation: ​[koʃi]; English: /ˈkoʊʃiː/ KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other a
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particula
Duality theory for distributive lattices
In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pair
Near sets
In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets, since they alw
Symmetry-protected topological order
Symmetry-protected topological (SPT) order is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap. To derive the results in a most-inva
I-adic topology
In commutative algebra, the mathematical study of commutative rings, adic topologies are a family of topologies on elements of a module, generalizing the p-adic topologies on the integers.
Pytkeev space
In mathematics, and especially topology, a Pytkeev space is a topological space that satisfies qualities more subtle than a convergence of a sequence. They are named after E. G. Pytkeev, who proved in
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a
Esakia duality
In mathematics, Esakia duality is the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an order-topological representation of Heytin
Cone (topology)
In topology, especially algebraic topology, the cone of a topological space is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. The cone of X i
Nikiel's conjecture
In mathematics, Nikiel's conjecture in general topology was a conjectural characterization of the continuous image of a compact total order. The conjecture was first formulated by in 1986. The conject
Perfect set
In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if , where denotes the set of all limit points of , also kn
Pointed space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space,
Quasi-open map
In topology a branch of mathematics, a quasi-open map or quasi-interior map is a function which has similar properties to continuous maps. However, continuous maps and quasi-open maps are not related.
Radial set
In mathematics, a subset of a linear space is radial at a given point if for every there exists a real such that for every Geometrically, this means is radial at if for every there is some (non-degene
Configuration space (mathematics)
In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point
O-minimal theory
In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) which is totally ordered by < is called an o-minimal structure if and only if every definable subset X ⊂ M
Nielsen theory
Nielsen theory is a branch of mathematical research with its origins in topological . Its central ideas were developed by Danish mathematician Jakob Nielsen, and bear his name. The theory developed in
Refinement (topology)
No description available.
Coincidence point
In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain having the same image. Formally, given two functions we say that a point x in X is a coin
Infinite loop space machine
In topology, a branch of mathematics, given a topological monoid X up to homotopy (in a nice way), an infinite loop space machine produces a group completion of X together with infinite loop space str
Path (topology)
In mathematics, a path in a topological space is a continuous function from the closed unit interval into Paths play an important role in the fields of topology and mathematical analysis. For example,
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection,
Imaging cycler microscopy
An imaging cycler microscope (ICM) is a fully automated (epi)fluorescence microscope which overcomes the spectral resolution limit resulting in parameter- and dimension-unlimited fluorescence imaging.
Topological abelian group
In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group.That is, a TAG is both a group and a topological space, the group operations are continuous, a
Extension topology
In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described
List of topology topics
In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, su
Quasitoric manifold
In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth -dimensional manifold is a quasitoric manifold if it admits
Totally bounded space
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered b
Cosmic space
In mathematics, particularly topology, a cosmic space is any topological space that is a continuous image of some separable metric space. Equivalently (for regular T1 spaces but not in general), a spa
In topology, a field within mathematics, desuspension is an operation inverse to suspension.
Generalised metric
In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field. In general, when we d
Lindelöf's lemma
In mathematics, Lindelöf's lemma is a simple but useful lemma in topology on the real line, named for the Finnish mathematician Ernst Leonard Lindelöf.
In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes,
Triangulation (topology)
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic
Weakly contractible
In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial.
Morita conjectures
The Morita conjectures in general topology are certain problems about normal spaces, now solved in the affirmative. The conjectures, formulated by Kiiti Morita in 1976, asked 1. * If is normal for ev
Fréchet distance
In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet.
Hurewicz space
In mathematics, a Hurewicz space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Hurewicz space is a space in which for every sequence of op
Secondary vector bundle structure
In mathematics, particularly differential topology, the secondary vector bundle structurerefers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smo
Compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space hav
Derived scheme
In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf either of simplicial commutative rings or of commutative ring spectra on X such that (1) the pair is a
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relat
Distortion (mathematics)
In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it
Topological complexity
In mathematics, topological complexity of a topological space X (also denoted by TC(X)) is a topological invariant closely connected to the motion planning problem, introduced by Michael Farber in 200
Domain (mathematical analysis)
In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rn or the complex
Whitehead's point-free geometry
In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereo
Homeomorphism group
In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group oper
Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are n
Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be con
Formal ball
In topology, a formal ball is an extension of the notion of ball to allow unbounded and negative radius. The concept of formal ball was introduced by Weihrauch and Schreiber in 1981 and the negative r
Nowhere continuous function
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f is a function from real numbers t
Kuranishi structure
In mathematics, especially in topology, a Kuranishi structure is a smooth analogue of scheme structure. If a topological space is endowed with a Kuranishi structure, then locally it can be identified
Support (mathematics)
In mathematics, the support of a real-valued function is the subset of the function domain containing the elements which are not mapped to zero. If the domain of is a topological space, then the suppo
Kuratowski's closure-complement problem
In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given st
Natural topology
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 defini
De Groot dual
In mathematics, in particular in topology, the de Groot dual (after Johannes de Groot) of a topology τ on a set X is the topology τ* whose closed sets are generated by compact saturated subsets of (X,
Condensed mathematics
Condensed mathematics is a theory developed by and Peter Scholze which aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry.
Adams resolution
In mathematics, specifically algebraic topology, there is a resolution analogous to free resolutions of spectra yielding a tool for constructing the Adams spectral sequence. Essentially, the idea is t
Handle decompositions of 3-manifolds
In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study.
Size function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane to the natural numbers, counting certain connected components of a topological space. T
Twist (mathematics)
In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon be composted of space curve , where is the arc length of , and the a unit normal vector, perpendicular at ea
Ribbon (mathematics)
In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by includes a curve given by a three-dimen
Banach fixed-point theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the exi
Monotonically normal space
In mathematics, specifically in the field of topology, a monotonically normal space is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesti
Puncture (topology)
In topology, puncturing a manifold is removing a finite set of points from that manifold. The set of points can be small as a single point. In this case, the manifold is known as once-punctured. With
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was disco
Lakes of Wada
In mathematics, the lakes of Wada (和田の湖, Wada no mizuumi) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundar
Velocity Energy-efficient and Link-aware Cluster-Tree (VELCT) is a cluster and tree-based topology management protocol for mobile wireless sensor networks (MWSNs).
Polytopological space
In general topology, a polytopological space consists of a set together with a family of topologies on that is linearly ordered by the inclusion relation ( is an arbitrary index set). It is usually as
Base change theorems
In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of
Quotient space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the or
Topological quantum computer
A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whos
Hutchinson metric
In mathematics, the Hutchinson metric otherwise known as Kantorovich metric is a function which measures "the discrepancy between two images for use in fractal image processing" and "can also be appli
Unit interval
In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital let
Listing number
In mathematics, a Listing number of a topological space is one of several topological invariants introduced by the 19th-century mathematician Johann Benedict Listing and later given this name by Charl
Nerve complex
In topology, the nerve complex of a set family is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov and now has many v
Atiyah–Jones conjecture
In mathematics, the Atiyah–Jones conjecture is a conjecture about the homology of the moduli spaces of instantons. The original form of the conjecture considered instantons over a 4 dimensional sphere
Lamination (topology)
In topology, a branch of mathematics, a lamination is a : * "topological space partitioned into subsets" * decoration (a structure or property at a point) of a manifold in which some subset of the m
Topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were
Topological rigidity
In the mathematical field of topology, a manifold M is called topologically rigid if every manifold homotopically equivalent to M is also homeomorphic to M.
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
Locally closed subset
In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if the following equivalent conditions are met: * E is an intersection of an open subset and a c
Direct sum of topological groups
In mathematics, a topological group is called the topological direct sum of two subgroups and if the map is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.
Waraszkiewicz spiral
In mathematics, Waraszkiewicz spirals are subsets of the plane introduced by Waraszkiewicz. Waraszkiewicz spirals give an example of an uncountable family of pairwise incomparable continua, meaning th
Antoine's necklace
In mathematics Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the cla
Circuit topology
The circuit topology of a folded linear polymer refers to the arrangement of its intra-molecular contacts. Examples of linear polymers with intra-molecular contacts are nucleic acids and proteins. Pro
Fixed-point index
In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed
Cauchy-continuous function
In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful pr
Toponomics is a discipline in systems biology, molecular cell biology, and histology concerning the study of the toponome of organisms. It is the field of study that purposes to decode the complete to
Selection principle
In mathematics, a selection principle is a rule assertingthe possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection pri
Pairwise Stone space
In mathematics and particularly in topology, pairwise Stone space is a bitopological space which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional. Pairwise Stone spaces are a bit
Tame topology
In mathematics, a tame topology is a hypothetical topology proposed by Alexander Grothendieck in his research program Esquisse d’un programme under the French name topologie modérée (moderate topology
Seven Bridges of Königsberg
The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topo
Topology optimization
Topology optimization (TO) is a mathematical method that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximizin
Caliber (mathematics)
In mathematics, the caliber or calibre of a topological space X is a cardinal κ such that for every set of κ nonempty open subsets of X there is some point of X contained in κ of these subsets. This c
Smash product
In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (X, x0) and (Y, y0) is the quotient of the product space X × Y und
Local flatness
In topology, a branch of mathematics, local flatness is smoothness condition that can be imposed on topological submanifolds. In the category of topological manifolds, locally flat submanifolds play a
Quasitopological space
In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a qu
Topological combinatorics
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics.
Partition of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval [0,1] such that for every point : * there is a neighbourhood of where all but a
Plateau (mathematics)
A plateau of a function is a part of its domain where the function has constant value. More formally, let U, V be topological spaces. A plateau for a function f: U → V is a path-connected set of point
Penrose triangle
The Penrose triangle, also known as the Penrose tribar, the impossible tribar, or the impossible triangle, is a triangular impossible object, an optical illusion consisting of an object which can be d
Poincaré–Miranda theorem
In mathematics, the Poincaré–Miranda theorem is a generalization of the intermediate value theorem, from a single function in a single dimension, to n functions in n dimensions. It says as follows: Co
Band sum
In geometric topology, a band sum of two n-dimensional knots K1 and K2 along an (n + 1)-dimensional 1-handle h called a band is an n-dimensional knot K such that: * There is an (n + 1)-dimensional 1-
Almgren–Pitts min-max theory
In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the effo
Regular space
In topology and related fields of mathematics, a topological space X is called a regular space if every closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods.
Trivial topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete o
Weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most
Bing–Borsuk conjecture
In mathematics, the Bing–Borsuk conjecture states that every -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and
Strong topology
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: * t
Derivative algebra (abstract algebra)
In abstract algebra, a derivative algebra is an algebraic structure of the signature where is a Boolean algebra and D is a unary operator, the derivative opera
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdor
Cellular space
A cellular space is a Hausdorff space that has the structure of a CW complex. * v * t * e
Separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given
Adjunction space
In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological
Topological degree theory
In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely c
In the field of functional analysis, DF-spaces, also written (DF)-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector sp