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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, topology (from the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
words , and ) is concerned with the properties of a
geometric object Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...
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 ga ...
deformations, such as
stretching Stretching is a form of physical exercise Exercise is any bodily activity that enhances or maintains physical fitness Physical fitness is a state of health Health is a state of physical, mental and social well-being Well-bein ...
, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
s, and, more generally,
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
s and
homotopies
homotopies
. A property that is invariant under such deformations is a
topological property In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
. Basic examples of topological properties are: the
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
, which allows distinguishing between a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
and a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
;
compactness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which allows distinguishing between a line and a circle;
connectedness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...
, who in the 17th century envisioned the and .
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Leonhard Euler
's
Seven Bridges of Königsberg The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographe ...
problem and
polyhedron formula
polyhedron formula
are arguably the field's first theorems. The term ''topology'' was introduced by
Johann Benedict Listing Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician. J. B. Listing was born in Frankfurt Frankfurt, officially Frankfurt am Main (; Hessian: ''Frangford am Maa'', " Frank ford on the Main"), is the ...
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 Kaliningrad ( ; rus, Калининград, p=kəlʲɪnʲɪnˈɡrat, links=y), until 1946 known as Königsberg (, ), is the largest city and the administrative centreAn administrative centre is a seat of regional administration or local gover ...

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 The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographe ...
problem led to the branch of mathematics known as
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Similarly, the
hairy ball theorem A hair whorl The hairy ball theorem of algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topologi ...
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 hair Hair is a protein filament that grows from follicles found in the dermis. Hair is one of the defining characteristics of mammals. The human body, apart from areas of glabrous skin, is covered in follicles wh ...
." 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
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. Another is
homotopy equivalence In topology, a branch of mathematics, two continuous function (topology), continuous functions from one topological space to another are called homotopic (from Ancient Greek, Greek ὁμός ''homós'' "same, similar" and τόπος ''tópos'' " ...
. 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 any bodily activity that enhances or maintains physical fitness Physical fitness is a state of health Health is a state of physical, mental and social well-being Well-being, also known as ''wellness'', ''prudential value ...
is to classify the uppercase letters of the
English alphabet The modern English alphabet is a Latin alphabet The Latin alphabet or Roman alphabet is the collection of letters originally used by the ancient Romans In historiography Historiography is the study of the methods of historian ...
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 Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language is a structured system of communication used by ...
Myriad A myriad (from Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), ...
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 class In mathematics, when the elements of some set (mathematics), set have a notion of equivalence (formalized ...

fundamental group
, is different on the supposedly differing classes. Letter topology has practical relevance in
stencil Stencilling produces an image or pattern by applying pigment to a surface under 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. The stencil is b ...

stencil
typography Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language is a structured system of communication used by humans, including ...

typography
. For instance, Braggadocio 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 Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Leonhard Euler
. His 1736 paper on the
Seven Bridges of Königsberg The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographe ...
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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
. 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 Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

Augustin-Louis Cauchy
, ,
Johann Benedict Listing Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician. J. B. Listing was born in Frankfurt Frankfurt, officially Frankfurt am Main (; Hessian: ''Frangford am Maa'', " Frank ford on the Main"), is the ...
,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
and
Enrico Betti Enrico Betti Glaoui (21 October 1823 – 11 August 1892) was an Italians, Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers. He worked also on the theory of equati ...

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 French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Repu ...
. In 1895, he published his ground-breaking paper on '' Analysis Situs'', which introduced the concepts now known as
homotopy In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric o ...

homotopy
and homology, which are now considered part of
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
. Unifying the work on function spaces of
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
,
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

Vito Volterra
, ,
Jacques Hadamard Jacques Salomon Hadamard ForMemRS (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis of the function . Hue represents the argument, brightness the magnitud ...
,
Giulio Ascoli Giulio Ascoli (20 January 1843, Trieste Trieste ( , ; sl, Trst ; german: Triest, ) is a city and a seaport File:PorticcioloCedas.jpg, The Porticciolo del Cedas port in Barcola near Trieste, a small local port A port is a maritime ...
and others,
Maurice FréchetMaurice may refer to: People *Saint Maurice Saint Maurice (also Moritz, Morris, or Mauritius; ) was the leader of the legendary Roman Theban Legion in the 3rd century, and one of the favorite and most widely venerated saints of that group. He w ...
introduced the
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
coined the term "topological space" and gave the definition for what is now called a
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

Hausdorff space
. Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by
Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a cou ...

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 Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
as part of his study of
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. For further developments, see
point-set topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
and algebraic topology.


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 tells how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, 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 remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

Cantor set
can be thought of as the same set with different topologies. Formally, let be a set and let be a
family In human society A society is a Social group, group of individuals involved in persistent Social relation, social interaction, or a large social group sharing the same spatial or social territory, typically subject to the same Politic ...
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 In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open set, open and closed set, closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but ...
), 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 British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
of .


Continuous functions and homeomorphisms

A
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

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 commun ...

one-to-one
and
onto In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ...

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. But the circle 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 British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
that is
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
to the Euclidean space of dimension .
Lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...

Lines
and
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
s, but not , are one-dimensional manifolds. Two-dimensional manifolds are also called
surfaces Water droplet lying on a damask. Surface tension">damask.html" ;"title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most gener ...
, although not all
surfaces Water droplet lying on a damask. Surface tension">damask.html" ;"title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most gener ...
are manifolds. Examples include the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the
Klein bottle in three-dimensional space In topology, a branch of mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; it is a two-dimensional manifold against which a system for determining a normal vec ...

Klein bottle
and
real projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s, which are sets equipped with a
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, that is, a family of
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
s, called ''open sets'', which is closed under finite
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...

intersection
s and (finite or infinite) s. The fundamental concepts of topology, such as '' continuity'', ''
compactness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

compactness
'', and ''
connectedness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'', 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 space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s 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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, real and complex
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s and
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
s. Having a metric simplifies many proofs.


Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
homeomorphism, though usually most classify up to homotopy equivalence. The most important of these invariants are
homotopy group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, homology, and
cohomology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. 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 for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
is again a free group.


Differential topology

Differential topology is the field dealing with
differentiable function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

differentiable function
s on
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
s. It is closely related to
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
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 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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
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 is non-orientable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical a ...
,
handle decompositionIn mathematics, a handle decomposition of an ''m''-manifold ''M'' is a union :\emptyset = M_ \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_ \subset M_m = M where each M_i is obtained from M_ by the attaching of i-handles. A handle d ...
s, local flatness, crumpling and the planar and higher-dimensional Jordan-Schönflies theorem, Schönflies theorem. In high-dimensional topology, characteristic classes 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/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 (Bernhard Riemann, Riemann surfaces are complex curves) – by the uniformization theorem every Conformal geometry, conformal class of Metric (mathematics), 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 one considers instead the lattice (order), lattice of open sets as the basic notion of the theory, while Grothendieck topology, Grothendieck topologies are structures defined on arbitrary category theory, categories that allow the definition of sheaf (mathematics), 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 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, 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. Topology is also used in evolutionary biology to represent the relationship between phenotype and genotype. 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 torus, toroidal). The main method used by topological data analysis is to: # Replace a set of data points with a family of simplicial complexes, 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, such as domain theory, are formalized using topology. In this context, Steve Vickers (computer scientist), Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean algebra (structure), Boolean 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, quantum field theory 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 and elementary units in materials. The compressive strength of Crumpling, 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 Fractal dimension, dimensionality of surface structures is the subject of interest with applications in multi-body physics. A topological quantum field theory (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, the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Simon Donaldson, Donaldson, Vaughan Jones, Jones, Edward Witten, Witten, and Maxim Kontsevich, Kontsevich have all won Fields Medals for work related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings. In cosmology, topology can be used to describe the overall shape of the universe. This area of research is commonly known as spacetime topology.


Robotics

The possible positions of a robot can be described by a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
called Configuration space (physics), 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

Disentanglement puzzle, 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 * Equivariant topology * 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 * List of publications in mathematics#Topology, 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, . * Nicolas Bourbaki, 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 spaces.) * Wacław Sierpiński, ''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 Topology, Mathematical structures