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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
deformations, such as stretching,
twisting Twist may refer to: In arts and entertainment Film, television, and stage * ''Twist'' (2003 film), a 2003 independent film loosely based on Charles Dickens's novel ''Oliver Twist'' * ''Twist'' (2021 film), a 2021 modern rendition of ''Olive ...
, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
endowed with a structure, called a '' topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
and a surface;
compactness 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 having no "punctures" or "missing endpoints", i ...
, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the and . Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term ''topology'' was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.


Motivation

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside. In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a
cowlick A cowlick is a section of human 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. The most common site of a human cowlick is in the crown, but they can show up anywhere. Th ...
." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the ''Bridges of Königsberg'', the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes. To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. Homeomorphism can be considered the most basic
topological equivalence In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object. An introductory
exercise Exercise is a body activity that enhances or maintains physical fitness and overall health and wellness. It is performed for various reasons, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic ...
is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends on the font used, and on whether the strokes making up the letters have some thickness or are ideal curves with no thickness. The figures here use the
sans-serif 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. Sans-serif typefaces tend to have less stroke width variation than seri ...
Myriad A myriad (from Ancient Greek grc, μυριάς, translit=myrias, label=none) is technically the number 10,000 (ten thousand); in that sense, the term is used in English almost exclusively for literal translations from Greek, Latin or Sinospher ...
font and are assumed to consist of ideal curves without thickness. Homotopy equivalence is a coarser relationship than homeomorphism; a homotopy equivalence class can contain several homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the "hole" part. Homeomorphism classes are: * no holes corresponding with C, G, I, J, L, M, N, S, U, V, W, and Z; * no holes and three tails corresponding with E, F, T, and Y; * no holes and four tails corresponding with X; * one hole and no tail corresponding with D and O; * one hole and one tail corresponding with P and Q; * one hole and two tails corresponding with A and R; * two holes and no tail corresponding with B; and * a bar with four tails corresponding with H and K; the "bar" on the ''K'' is almost too short to see. Homotopy classes are larger, because the tails can be squished down to a point. They are: * one hole, * two holes, and * no holes. To classify the letters correctly, we must show that two letters in the same class are equivalent and two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by selecting points and showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
, is different on the supposedly differing classes. Letter topology has practical relevance in
stencil Stencilling produces an image or pattern on a surface, by applying pigment to a surface through an intermediate object, with designed holes in the intermediate object, to create a pattern or image on a surface, by allowing the pigment to reach ...
typography. For instance,
Braggadocio Braggadocio may refer to: *Braggadocchio, a fictional character in the epic poem ''The Faerie Queene'' *A braggart or empty boasting *Braggadocio (rap), a type of rapping *Braggadocio (typeface), a typeface *Braggadocio, Missouri Braggadocio is ...
font stencils are made of one connected piece of material.


History

Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realized the importance of the ''edges'' of a polyhedron. This led to his polyhedron formula, (where , , and respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology. Further contributions were made by
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 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. He ...
, Ludwig Schläfli, Johann Benedict Listing,
Bernhard Riemann 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 the field of real analysis, he is mostly known for the first rig ...
and Enrico Betti.Richeson (2008) Listing introduced the term "Topologie" in ''Vorstudien zur Topologie'', written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print. The English form "topology" was used in 1883 in Listing's obituary in the journal ''Nature'' to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". Their work was corrected, consolidated and greatly extended by
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
. In 1895, he published his ground-breaking paper on '' Analysis Situs'', which introduced the concepts now known as homotopy and
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others,
Maurice Fréchet Maurice may refer to: People *Saint Maurice (died 287), Roman legionary and Christian martyr *Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor *Maurice (bishop of London) (died 1107), Lord Chancellor and Lo ...
introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
. For further developments, see point-set topology and algebraic topology. The 2022
Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".


Concepts


Topologies on sets

The term ''topology'' also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, and the
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...
can be thought of as the same set with different topologies. Formally, let be a set and let be a family of subsets of . Then is called a topology on if: # Both the empty set and are elements of . # Any union of elements of is an element of . # Any intersection of finitely many elements of is an element of . If is a topology on , then the pair is called a topological space. The notation may be used to denote a set endowed with the particular topology . By definition, every topology is a -system. The members of are called ''open sets'' in . A subset of is said to be closed if its complement is in (that is, its complement is open). A subset of may be open, closed, both (a clopen set), or neither. The empty set and itself are always both closed and open. An open subset of which contains a point is called a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of .


Continuous functions and homeomorphisms

A function or map from one topological space to another is called ''continuous'' if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is
one-to-one One-to-one or one to one may refer to: Mathematics and communication *One-to-one function, also called an injective function *One-to-one correspondence, also called a bijective function *One-to-one (communication), the act of an individual comm ...
and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to the doughnut.


Manifolds

While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A ''manifold'' is a topological space that resembles Euclidean space near each point. More precisely, each point of an -dimensional manifold has a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
that is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the Euclidean space of dimension .
Lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces, although not all surfaces are manifolds. Examples include the
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...
, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the
Klein bottle 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. Informally, it is a o ...
and real projective plane, which cannot (that is, all their realizations are surfaces that are not manifolds).


Topics


General topology

General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s, called ''open sets'', which is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
under finite
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
s and (finite or infinite) unions. The fundamental concepts of topology, such as '' continuity'', ''
compactness 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 having no "punctures" or "missing endpoints", i ...
'', and '' connectedness'', can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words ''nearby'', ''arbitrarily small'', and ''far apart'' can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected. Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a ''metric''. In a metric space, an open set is a union of open disks, where an open disk of radius centered at is the set of all points whose distance to is less than . Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, real and complex vector spaces and Euclidean spaces. Having a metric simplifies many proofs.


Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal is to find algebraic invariants that
classify Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood. Classification is the grouping of related facts into classes. It may also refer to: Business, organizat ...
topological spaces
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
homeomorphism, though usually most classify up to homotopy equivalence. The most important of these invariants are
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s, homology, and cohomology. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.


Differential topology

Differential topology is the field dealing with differentiable functions on
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s. It is closely related to
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and
Riemannian curvature 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 way used to express the curvature of Riemannian manifolds. ...
are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.


Geometric topology

Geometric topology is a branch of topology that primarily focuses on low-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology. Some examples of topics in geometric topology are orientability, handle decompositions,
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 role similar to that of embedded submanifol ...
, crumpling and the planar and higher-dimensional Schönflies theorem. In high-dimensional topology,
characteristic classes In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
are a basic invariant, and surgery theory is a key theory. Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
/spherical, zero curvature/flat, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries. 2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.


Generalizations

Occasionally, one needs to use the tools of topology but a "set of points" is not available. In
pointless topology In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this appr ...
one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of general cohomology theories.


Applications


Biology

Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and
knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings.
Knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower
electrophoresis Electrophoresis, from Ancient Greek ἤλεκτρον (ḗlektron, "amber") and φόρησις (phórēsis, "the act of bearing"), is the motion of dispersed particles relative to a fluid under the influence of a spatially uniform electric fie ...
. Topology is also used in evolutionary biology to represent the relationship between phenotype and
genotype The genotype of an organism is its complete set of genetic material. Genotype can also be used to refer to the alleles or variants an individual carries in a particular gene or genetic location. The number of alleles an individual can have in a ...
. Phenotypic forms that appear quite different can be separated by only a few mutations depending on how genetic changes map to phenotypic changes during development. In neuroscience, topological quantities like the Euler characteristic and Betti number have been used to measure the complexity of patterns of activity in neural networks.


Computer science

Topological data analysis uses techniques from algebraic topology to determine the large scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysis is to: # Replace a set of data points with a family of
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
es, indexed by a proximity parameter. # Analyse these topological complexes via algebraic topology – specifically, via the theory of persistent homology. # Encode the persistent homology of a data set in the form of a parameterized version of a Betti number, which is called a barcode. Several branches of
programming language semantics In programming language theory, semantics is the rigorous mathematical study of the meaning of programming languages. Semantics assigns computational meaning to valid strings in a programming language syntax. Semantics describes the processe ...
, such as domain theory, are formalized using topology. In this context, Steve Vickers, building on work by
Samson Abramsky Samson Abramsky (born 12 March 1953) is Professor of Computer Science at University College London. He was previously the Christopher Strachey Professor of Computing at the University of Oxford, from 2000 to 2021. He has made contributions to t ...
and
Michael B. Smyth Michael may refer to: People * Michael (given name), a given name * Michael (surname), including a list of people with the surname Michael Given name "Michael" * Michael (archangel), ''first'' of God's archangels in the Jewish, Christian an ...
, characterizes topological spaces as
Boolean Any kind of logic, function, expression, or theory based on the work of George Boole is considered Boolean. Related to this, "Boolean" may refer to: * Boolean data type, a form of data with only two possible values (usually "true" and "false" ...
or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.


Physics

Topology is relevant to physics in areas such as
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
and physical cosmology. The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science. Electrical and mechanical properties depend on the arrangement and network structures of
molecules A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
and elementary units in materials. The
compressive strength In mechanics, compressive strength or compression strength is the capacity of a material or structure to withstand loads tending to reduce size (as opposed to tensile strength which withstands loads tending to elongate). In other words, compre ...
of crumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space. Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics. A
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 invented by physicists, they are also of mathem ...
(or topological field theory or TQFT) is a quantum field theory that computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things,
knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
, the theory of four-manifolds in algebraic topology, and to the theory of
moduli spaces In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
in algebraic geometry.
Donaldson Donaldson is a Scottish and Irish patronymic surname meaning "son of Donald". It is a simpler Anglicized variant for the name MacDonald. Notable people with the surname include: __NOTOC__ A * Alastair Donaldson (1955–2013), Scottish musician ...
,
Jones Jones may refer to: People *Jones (surname), a common Welsh and English surname *List of people with surname Jones * Jones (singer), a British singer-songwriter Arts and entertainment * Jones (''Animal Farm''), a human character in George Orwell ...
, Witten, and
Kontsevich Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques a ...
have all won
Fields Medal 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. The name of the award ho ...
s for work related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in
string theory 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. String theory describes how these strings propagate through space and interac ...
, as different manifolds can sustain different kinds of strings. In cosmology, topology can be used to describe the overall
shape of the universe The shape of the universe, in physical cosmology, is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes gen ...
. This area of research is commonly known as spacetime topology. In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect, and then generalized in other areas of physics, for instance in photonics by F.D.M Haldane.


Robotics

The possible positions of a robot can be described by a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
called configuration space. In the area of motion planning, one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose.


Games and puzzles

Tanglement puzzles are based on topological aspects of the puzzle's shapes and components.


Fiber art

In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order which surrounds each piece and traverses each edge only once. This process is an application of the Eulerian path.


See also

*
Characterizations of the category of topological spaces In the mathematical field of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a t ...
* Equivariant topology * List of algebraic topology topics * List of examples in general topology * List of general topology topics *
List of geometric topology topics This is a list of geometric topology topics, by Wikipedia page. See also: *topology glossary *List of topology topics *List of general topology topics *List of algebraic topology topics * Publications in topology Low-dimensional topology Knot the ...
* List of topology topics * Publications in topology * Topoisomer * Topology glossary * Topological Galois theory * Topological geometry * Topological order


References


Citations


Bibliography

* * *


Further reading

* Ryszard Engelking, ''General Topology'', Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, . * Bourbaki; ''Elements of Mathematics: General Topology'', Addison–Wesley (1966). * * * (Provides a well motivated, geometric account of general topology, and shows the use of groupoids in discussing van Kampen's theorem, covering spaces, and
orbit space In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
s.) *
Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and to ...
, ''General Topology'', Dover Publications, 2000, * (Provides a popular introduction to topology and geometry) *


External links

*
Elementary Topology: A First Course
Viro, Ivanov, Netsvetaev, Kharlamov. *
The Topological Zoo
at The Geometry Center.
Topology Atlas


Aisling McCluskey and Brian McMaster, Topology Atlas.
Topology Glossary

Moscow 1935: Topology moving towards America
a historical essay by Hassler Whitney. {{Authority control Mathematical structures