162 relations: Abstract algebra, Algebraic geometry, Algebraic topology, Analysis Situs (paper), Augustin-Louis Cauchy, Bernhard Riemann, Betti number, Braggadocio (typeface), Calabi–Yau manifold, Calculus, Cantor set, Category theory, Cesare Arzelà, Characteristic class, Circle, Classification theorem, Clopen set, Closed set, Cohomology, Compact space, Complex plane, Condensed matter physics, Configuration space (physics), Connectedness, Contact mechanics, Continuous function, Covering space, Cowlick, Cross-cap, Crumpling, Deformation theory, Differentiable function, Differentiable manifold, Differential geometry, Differential topology, Disentanglement puzzle, Edward Witten, Electrophoresis, English alphabet, Enrico Betti, Equivariant topology, Euclidean space, Euler characteristic, Eulerian path, Evolutionary biology, Exercise (mathematics), Family of sets, Felix Hausdorff, Fields Medal, Font, ..., Fourier series, Fractal dimension, Free group, Function (mathematics), Fundamental group, General topology, Genotype, Genus (mathematics), Geometric topology, Geometrization conjecture, Geometry, Geometry Center, Georg Cantor, Giulio Ascoli, Glossary of topology, Gottfried Wilhelm Leibniz, Graph theory, Greek language, Grothendieck topology, Group action, Hairy ball theorem, Handle decomposition, Hassler Whitney, Hausdorff space, Henri Poincaré, Homeomorphism, Homology (mathematics), Homotopy, Homotopy group, Injective function, Invariant (mathematics), Jacques Hadamard, Johann Benedict Listing, Kaliningrad, Kazimierz Kuratowski, Klein bottle, Knot theory, Lattice (order), Lemniscate, Leonhard Euler, Limit of a sequence, Line (geometry), List of algebraic topology topics, List of examples in general topology, List of general topology topics, List of geometric topology topics, List of topology topics, Local flatness, Ludwig Schläfli, Manifold, Materials science, Mathematics, Maurice René Fréchet, Maxim Kontsevich, Mechanical engineering, Metric space, Moduli space, Molecule, Motion planning, Myriad (typeface), Nature (journal), Neighbourhood (mathematics), Nicolas Bourbaki, Open set, Orientability, Persistent homology, Phenotype, Physical cosmology, Plane (geometry), Pointless topology, Polyhedron, Projective plane, Quantum field theory, Real line, Real number, Real projective plane, Riemann curvature tensor, Ryszard Engelking, Sans-serif, Schoenflies problem, Seifert–van Kampen theorem, Set theory, Seven Bridges of Königsberg, Sheaf (mathematics), Simon Donaldson, Simplicial complex, Smooth structure, Spacetime topology, Sphere, Springer Science+Business Media, Stencil, String theory, Surface (topology), Surgery theory, Surjective function, The Spectator, Topoisomer, Topological data analysis, Topological geometry, Topological order, Topological property, Topological space, Torus, Typography, Uniformization theorem, Unknot, Up to, Vaughan Jones, Vector field, Vito Volterra, Wacław Sierpiński, 4-manifold. Expand index (112 more) » « Shrink index
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
"Analysis Situs" is a seminal mathematics paper that Henri Poincaré published in 1895.
Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.
Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.
Braggadocio is a geometrically constructed sans-serif stencil typeface designed by W.A. Woolley in 1930 for the Monotype Corporation.
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics.
Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
Cesare Arzelà (6 March 1847 – 15 March 1912) was an Italian mathematician who taught at the University of Bologna and is recognized for his contributions in the theory of functions, particularly for his characterization of sequences of continuous functions, generalizing the one given earlier by Giulio Ascoli in the Arzelà-Ascoli theorem.
In mathematics, a characteristic class is a way of associating to each principal bundle X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" — and whether it possesses sections.
A circle is a simple closed shape.
In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?".
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed.
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter.
In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system.
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece".
Contact mechanics is the study of the deformation of solids that touch each other at one or more points.
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.
A cowlick is a section of hair that stands straight up or lies at an angle at odds with the style in which the rest of an individual's hair is worn.
In mathematics, a cross-cap is a two-dimensional surface in 3-space that is one-sided and the continuous image of a Möbius strip that intersects itself in an interval.
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 random network of ridges and facets with variable density.
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities.
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.
Disentanglement puzzles (also called tanglement puzzles, tavern puzzles or topological puzzles) are a type of mechanical puzzle that involves disentangling one piece or set of pieces from another piece or set of pieces.
Edward Witten (born August 26, 1951) is an American theoretical physicist and professor of mathematical physics at the Institute for Advanced Study in Princeton, New Jersey.
Electrophoresis (from the Greek "Ηλεκτροφόρηση" meaning "to bear electrons") is the motion of dispersed particles relative to a fluid under the influence of a spatially uniform electric field.
The modern English alphabet is a Latin alphabet consisting of 26 letters, each having an uppercase and a lowercase form: The same letters constitute the ISO basic Latin alphabet.
Enrico Betti Glaoui (21 October 1823 – 11 August 1892) was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers.
In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries.
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph which visits every edge exactly once.
Evolutionary biology is the subfield of biology that studies the evolutionary processes that produced the diversity of life on Earth, starting from a single common ancestor.
A mathematical exercise is a routine application of algebra or other mathematics to a stated challenge.
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years.
In metal typesetting, a font was a particular size, weight and style of a typeface.
In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured.
In mathematics, the free group FS over a given set S consists of all expressions (a.k.a. words, or terms) that can be built from members of S, considering two expressions different unless their equality follows from the group axioms (e.g. st.
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.
The genotype is the part of the genetic makeup of a cell, and therefore of an organism or individual, which determines one of its characteristics (phenotype).
In mathematics, genus (plural genera) has a few different, but closely related, meanings.
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
The Geometry Center was a mathematics research and education center at the University of Minnesota.
Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.
Giulio Ascoli (20 January 1843, Trieste – 12 July 1896, Milan) was a Jewish-Italian mathematician.
This is a glossary of some terms used in the branch of mathematics known as topology.
Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
Greek (Modern Greek: ελληνικά, elliniká, "Greek", ελληνική γλώσσα, ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean and the Black Sea.
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres.
In mathematics, a handle decomposition of an m-manifold M is a union where each M_i is obtained from M_ by the attaching of i-handles.
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician.
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.
Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.
In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.
Jacques Salomon Hadamard ForMemRS (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.
Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician.
Kaliningrad (p; former German name: Königsberg; Yiddish: קעניגסבערג, Kenigsberg; r; Old Prussian: Twangste, Kunnegsgarbs, Knigsberg; Polish: Królewiec) is a city in the administrative centre of Kaliningrad Oblast, a Russian exclave between Poland and Lithuania on the Baltic Sea.
Kazimierz Kuratowski (Polish pronunciation:, 2 February 1896 – 18 June 1980) was a Polish mathematician and logician.
In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.
In topology, knot theory is the study of mathematical knots.
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves.
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
This is a list of algebraic topology topics, by Wikipedia page.
This is a list of useful examples in general topology, a field of mathematics.
This is a list of general topology topics, by Wikipedia page.
This is a list of geometric topology topics, by Wikipedia page.
This is a list of topology topics, by Wikipedia page.
In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension.
Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
The interdisciplinary field of materials science, also commonly termed materials science and engineering is the design and discovery of new materials, particularly solids.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
Maurice Fréchet (2 September 1878 – 4 June 1973) was a French mathematician.
Maxim Lvovich Kontsevich (Макси́м Льво́вич Конце́вич;; born 25 August 1964) is a Russian and French mathematician.
Mechanical engineering is the discipline that applies engineering, physics, engineering mathematics, and materials science principles to design, analyze, manufacture, and maintain mechanical systems.
In mathematics, a metric space is a set for which distances between all members of the set are defined.
In algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.
A molecule is an electrically neutral group of two or more atoms held together by chemical bonds.
Motion planning (also known as the navigation problem or the piano mover's problem) is a term used in robotics for the process of breaking down a desired movement task into discrete motions that satisfy movement constraints and possibly optimize some aspect of the movement.
Myriad is a humanist sans-serif typeface designed by Robert Slimbach and Carol Twombly for Adobe Systems.
Nature is a British multidisciplinary scientific journal, first published on 4 November 1869.
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.
In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.
Persistent homology is a method for computing topological features of a space at different spatial resolutions.
A phenotype is the composite of an organism's observable characteristics or traits, such as its morphology, development, biochemical or physiological properties, behavior, and products of behavior (such as a bird's nest).
Physical cosmology is the study of the largest-scale structures and dynamics of the Universe and is concerned with fundamental questions about its origin, structure, evolution, and ultimate fate.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points.
In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.
In mathematics, a projective plane is a geometric structure that extends the concept of a plane.
In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.
In mathematics, the real line, or real number line is the line whose points are the real numbers.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.
Ryszard Engelking (born 1935 in Sosnowiec) is a Polish mathematician.
In typography and lettering, a sans-serif, sans serif, gothic, or simply sans letterform is one that does not have extending features called "serifs" at the end of strokes.
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies.
In mathematics, the Seifert–van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X in terms of the fundamental groups of two open, path-connected subspaces that cover X. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
The Seven Bridges of Königsberg is a historically notable problem in mathematics.
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
Sir Simon Kirwan Donaldson FRS (born 20 August 1957), is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory.
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function.
Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity.
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Stencilling produces an image or pattern by applying pigment to a surface over an intermediate object with designed gaps in it which create the pattern or image by only allowing the pigment to reach some parts of the surface.
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by.
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
The Spectator is a weekly British magazine on politics, culture, and current affairs.
Topoisomers or topological isomers are molecules with the same chemical formula and stereochemical bond connectivities but different topologies.
In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology.
Topological geometry deals with incidence structures consisting of a point set P and a family \mathfrak of subsets of P called lines or circles etc.
In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter).
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
Typography is the art and technique of arranging type to make written language legible, readable, and appealing when displayed.
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.
The unknot arises in the mathematical theory of knots.
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
Sir Vaughan Frederick Randal Jones (born 31 December 1952) is a New Zealand and American mathematician, known for his work on von Neumann algebras and knot polynomials.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
Vito Volterra (3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis.
Wacław Franciszek Sierpiński (14 March 1882 – 21 October 1969) was a Polish mathematician.
In mathematics, a 4-manifold is a 4-dimensional topological manifold.