View source for Topology

{{Short description|Branch of mathematics}}
{{Other uses}}
{{Distinguish|Topography|Typology}}
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[[File:Torus knot 2.stl|thumb|upright=1.5|A three-dimensional model of a [[Figure-eight knot (mathematics)|figure-eight knot]]. The figure-eight knot is a [[prime knot]] and has an [[Alexander–Briggs notation]] of 4<sub>1</sub>.]]

'''Topology''' (from the [[Greek language|Greek]] words {{Langx|grc|[[wikt:τόπος|τόπος]]|lit=place, location|label=none}}, and {{Langx|el|λόγος|lit=study|label=none}}) is the branch of [[mathematics]] concerned with the properties of a [[Mathematical object|geometric object]] that are preserved under [[Continuous function|continuous]] [[Deformation theory|deformations]], such as [[Stretch factor|stretching]], [[Torsion (mechanics)|twisting]], crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

A [[topological space]] is a [[Set (mathematics)|set]] endowed with a structure, called a ''[[Topology (structure)|topology]]'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of [[List of continuity-related mathematical topics|continuity]]. [[Euclidean space]]s, and, more generally, [[metric space]]s are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are [[homeomorphism]]s and [[Homotopy|homotopies]]. A property that is invariant under such deformations is a [[topological property]]. The following are basic examples of topological properties: the [[Lebesgue covering dimension|dimension]], which allows distinguishing between a [[Line (geometry)|line]] and a [[Surface (mathematics)|surface]]; [[Compact space|compactness]], 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 Wilhelm Leibniz]], who in the 17th century envisioned the {{lang|la|geometria situs}} and {{lang|la|analysis situs}}. [[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==
[[Image:Möbius strip.jpg|thumb|240px|[[Möbius strip]]s, which have only one surface and one edge, are a kind of object studied in topology.]]
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.<ref>{{cite journal |last1=Euler |first1=Leonhard |author1-link=Leonhard Euler |title=Solutio problematis ad geometriam situs pertinensis |journal=Commentarii Academiae Scientarum Imperialis Petropolitanae |date=1736 |volume=8 |pages=128–140 |url=https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler.html |archive-date=29 June 2025 |access-date=29 June 2025 |archive-url=https://web.archive.org/web/20250629133524/https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler.html |url-status=live }}</ref> 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]].<ref>{{cite journal |last1=Shields |first1=Rob |title=Cultural Topology: The Seven Bridges of Königsburg, 1736 |journal=Theory, Culture & Society |date=1 July 2012 |volume=29 |issue=4–5 |pages=43–57 |doi=10.1177/0263276412451161 |url=https://journals.sagepub.com/doi/10.1177/0263276412451161 |language=EN |issn=0263-2764 |archive-date=29 June 2025 |access-date=29 June 2025 |archive-url=https://web.archive.org/web/20250629141615/https://journals.sagepub.com/doi/10.1177/0263276412451161 |url-status=live |url-access=subscription }}</ref>

Similarly, the [[hairy ball theorem]] of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a [[cowlick]]."<ref>{{cite journal |last1=Eisenberg |first1=Murray |last2=Guy |first2=Robert |title=A Proof of the Hairy Ball Theorem |journal=The American Mathematical Monthly |date=1 August 1979 |volume=86 |issue=7 |pages=571–574 |doi=10.1080/00029890.1979.11994857 |issn=0002-9890}}</ref> 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 [[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 do 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.

[[File:Topology joke.jpg|thumb|right|240px|A continuous transformation can turn a coffee mug into a donut.<br/>Ceramic model by Keenan Crane and [[Henry Segerman]].]]
Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A famous example, known as the "Topologist's Breakfast", is that a topologist cannot distinguish a coffee mug from a doughnut.<ref>{{cite web |last1=Roux |first1=Mariëtte Le |last2=Hood |first2=Marlowe |title=When is a coffee mug a donut? Topology explains it |url=https://phys.org/news/2016-10-coffee-donut-topology.html |website=phys.org |access-date=29 June 2025 |language=en |date=4 October 2016 |archive-date=29 June 2025 |archive-url=https://web.archive.org/web/20250629141832/https://phys.org/news/2016-10-coffee-donut-topology.html |url-status=live }}</ref> A pliable [[torus]] (shaped like a doughnut) can be reshaped to a coffee mug by creating a dimple and progressively enlarging it while shrinking the central hole into the mug's handle.<ref>{{cite book|url=https://books.google.com/books?id=SHBj2oaSALoC&pg=PA204|title=Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems|last1=Hubbard|first1=John H.|last2=West|first2=Beverly H.|publisher=Springer|year=1995|isbn=978-0-387-94377-0|series=Texts in Applied Mathematics|volume=18|page=204}}</ref>

Homeomorphism can be considered the most basic [[Homeomorphism|topological equivalence]]. 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.

==History==
[[Image:Konigsberg bridges.png|thumb|right|240px|The [[Seven Bridges of Königsberg]] was a problem solved by Euler.]]
{{See also|History of the separation axioms}}

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.<ref name=Croom7>{{harvnb|Croom|1989|page=7}}</ref> 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.<ref name=Croom7 /> 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 [[Euler's polyhedron formula|polyhedron formula]], {{math|1=''V'' − ''E'' + ''F'' = 2}} (where {{mvar|V}}, {{mvar|E}}, and {{mvar|F}} 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.<ref>{{harvnb|Richeson|2008|page=63}}; {{harvnb|Aleksandrov|1969|page=204}}</ref>

Further contributions were made by [[Augustin-Louis Cauchy]], [[Ludwig Schläfli]], [[Johann Benedict Listing]], [[Bernhard Riemann]] and [[Enrico Betti]].<ref name="Richeson">Richeson (2008)</ref> 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.<ref>Listing, Johann Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p.&nbsp;67, 1848</ref> The English form "topology" was used in 1883 in Listing's obituary in the journal [[Nature (journal)|''Nature'']] to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated".<ref>{{cite journal|doi=10.1038/027316a0|title=Johann Benedict Listing (obituary)|date=1 February 1883|last1=Tait|first1=Peter Guthrie|journal=Nature|volume=27|issue=692|pages=316–317|bibcode=1883Natur..27..316P|doi-access=free}}</ref>

Their work was corrected, consolidated and greatly extended by [[Henri Poincaré]]. In 1895, he published his ground-breaking paper on ''[[Analysis Situs (paper)|Analysis Situs]]'', which introduced the concepts now known as [[homotopy]] and [[Homology (mathematics)|homology]], which are now considered part of [[algebraic topology]].<ref name="Richeson"/>

{| class="wikitable sortable"
|+ Topological characteristics of closed 2-manifolds<ref name="Richeson" />
|-
! rowspan=2 | Manifold !! rowspan=2 |[[Euler characteristic|Euler number]]!! rowspan="2" |[[Orientability]]!! colspan="3" |[[Betti number]]s!! rowspan="2" |[[Homology (mathematics)|Torsion coefficient]]<br/>(1-dim)
|-
! style="width:3em;" | ''b''<sub>0</sub>!! ''b''<sub>1</sub>!! style="width:3em;" | ''b''<sub>2</sub>
|-
| [[Sphere]] || 2 || {{Yes2|Orientable}} || 1 || 0 || 1 || {{CNone|none}}
|-
| [[Torus]] || 0 || {{Yes2|Orientable}} || 1 || 2 || 1 || {{CNone|none}}
|-
| 2-holed torus || −2 || {{Yes2|Orientable}} || 1 || 4 || 1 || {{CNone|none}}
|-
| {{mvar|g}}-holed torus ([[Genus (topology)|genus]] {{mvar|g}}) || {{math|2 − 2''g''}} || {{Yes2|Orientable}} || 1 || {{math|2''g''}} || 1 || {{CNone|none}}
|-
| [[Projective plane]] || 1 || {{No2|Non-orientable}} || 1 || 0 || 0 || 2
|-
| [[Klein bottle]] || 0 || {{No2|Non-orientable}} || 1 || 1 || 0 || 2
|-
| Sphere with {{mvar|c}} [[cross-cap]]s ({{math|''c'' > 0}}) || {{math|2 − ''c''}} || {{No2|Non-orientable}} || 1 || {{math|''c'' − 1}} || 0 || 2
|-
| 2-Manifold with {{mvar|g}} holes<br />and {{mvar|c}} cross-caps ({{math|''c'' > 0}}) || {{math|2 − (2''g'' + ''c'')}} || {{No2|Non-orientable}} || 1 || {{math|(2''g'' + ''c'') − 1}} || 0 || 2
|}

The development of topology in the 20th century was marked by significant advances in both foundational theory and its application to other fields of mathematics. Unifying the work on function spaces of [[Georg Cantor]], [[Vito Volterra]], [[Cesare Arzelà]], [[Jacques Hadamard]], [[Giulio Ascoli]] and others, [[Maurice Fréchet]] introduced the [[metric space]] in 1906.<ref>{{cite book |last=Fréchet |first=Maurice |title=Sur quelques points du calcul fonctionnel |year=1906 |oclc=8897542 }}</ref> 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 defined what is now called a [[Hausdorff space]].<ref>Hausdorff, Felix, "Grundzüge der Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)</ref> Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by [[Kazimierz Kuratowski]].<ref>{{harvnb|Croom|1989|page=129}}</ref>

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]]. For further developments, see [[point-set topology]] and algebraic topology.

The 2022 [[Abel Prize]] was awarded to [[Dennis Sullivan]] "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".<ref name="Abel-2022">{{cite web |url=https://abelprize.no/citation/citation-dennis-parnell-sullivan |title=Prize winner 2022 |publisher=The Norwegian Academy of Science and Letters |access-date=23 March 2022 |archive-date=14 April 2022 |archive-url=https://web.archive.org/web/20220414033756/https://abelprize.no/citation/citation-dennis-parnell-sullivan |url-status=live }}</ref>

==Concepts==

===Topologies on sets===
{{Main|Topological space}}

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]], the [[complex plane]], and the [[Cantor set]] can be thought of as the same set with different topologies.

Formally, let {{mvar|X}} be a set and let {{mvar|τ}} be a [[family of sets|family]] of subsets of {{mvar|X}}. Then {{mvar|τ}} is called a topology on {{mvar|X}} if:

# Both the empty set and {{mvar|X}} are elements of {{mvar|τ}}.
# Any union of elements of {{mvar|τ}} is an element of {{mvar|τ}}.
# Any intersection of finitely many elements of {{mvar|τ}} is an element of {{mvar|τ}}.

If {{mvar|τ}} is a topology on {{mvar|X}}, then the pair {{math|(''X'', ''τ'')}} is called a topological space. The notation {{math|''X''<sub>''τ''</sub>}} may be used to denote a set {{mvar|X}} endowed with the particular topology {{mvar|τ}}. By definition, every topology is a [[Pi-system|{{pi}}-system]].

The members of {{mvar|τ}} are called ''open sets'' in {{mvar|X}}. A subset of {{mvar|X}} is said to be closed if its complement is in {{mvar|τ}} (that is, its complement is open). A subset of {{mvar|X}} may be open, closed, both (a [[clopen set]]), or neither. The empty set and {{mvar|X}} itself are always both closed and open. An open subset of {{mvar|X}} which contains a point {{mvar|x}} is called an open [[neighborhood (mathematics)|neighborhood]] of {{mvar|x}}.

===Continuous functions and homeomorphisms===
{{anchor|mug-and-doughnut}}<!-- used in [[Mathematical joke]] -->
{{multiple image
 | width = 200
 | image1 = Mug and Torus morph.gif
 | image2 = Spot the cow.gif
 | footer = A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and of a cow [[plushie]] into a sphere
}}
{{Main|Continuous function|homeomorphism}}

A [[function (mathematics)|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 number]]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]]. If a continuous function is [[injective function|one-to-one]] and [[surjective function|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===
{{Main|Manifold}}
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 {{mvar|n}}-dimensional manifold has a [[neighborhood (mathematics)|neighborhood]] that is [[homeomorphic]] to the Euclidean space of dimension {{mvar|n}}. [[Line (geometry)|Lines]] and [[circle]]s, but not [[Lemniscate|figure eights]], are one-dimensional manifolds. Two-dimensional manifolds are also called [[Surface (topology)|surfaces]]. Examples include the [[Plane (geometry)|plane]], the [[sphere]], and the [[torus]], which can all be realized  in three dimensions without self-intersection, and the [[Klein bottle]] and [[real projective plane]], which cannot.

==Topics==

===General topology===
{{Main|General topology}}
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.<ref>Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.</ref><ref>Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.</ref> 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]]s, which are sets equipped with a [[topology (structure)|topology]], that is, a family of [[subset]]s, called ''open sets'', which is [[closure (mathematics)|closed]] under finite [[set intersection|intersection]]s and (finite or infinite) [[set union|union]]s. The fundamental concepts of topology, such as ''[[continuity (mathematics)|continuity]]'', ''[[compactness]]'', 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 set. 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]]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 {{mvar|r}} centered at {{mvar|x}} is the set of all points whose distance to {{mvar|x}} is less than {{mvar|r}}. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the [[real line]], the [[complex plane]], real and complex [[vector space]]s and [[Euclidean space]]s. Having a metric simplifies many proofs.

===Algebraic topology===
{{Main|Algebraic topology}}
Algebraic topology is a branch of mathematics that uses tools from [[algebra]] to study topological spaces.<ref>Allen Hatcher, [https://pi.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic topology.''] {{Webarchive|url=https://web.archive.org/web/20120206155217/http://www.math.cornell.edu/~hatcher/AT/ATpage.html |date=6 February 2012 }} (2002) Cambridge University Press, xii+544 pp.&nbsp;{{isbn|0-521-79160-X|0-521-79540-0}}.</ref> The basic goal is to find algebraic invariants that [[classification theorem|classify]] topological spaces [[up to]] homeomorphism, or more commonly up to homotopy equivalence.

The most important of these invariants are [[homotopy group]]s, [[Homology_(mathematics)|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===
{{Main|Differential topology}}
Differential topology is the field dealing with [[differentiable function]]s on [[differentiable manifold]]s.<ref>{{cite book | first = John M. | last = Lee | year = 2006 | title = Introduction to Smooth Manifolds | publisher = Springer-Verlag | isbn = 978-0-387-95448-6}}</ref> It is closely related to [[differential geometry]] 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 [[Deformation theory|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{{snd}}that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

===Geometric topology===
{{Main|Geometric topology}}

Geometric topology is a branch of topology that primarily focuses on low-dimensional [[manifold]]s (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.<ref>R. B. Sher and [[R. J. Daverman]] (2002), ''Handbook of Geometric Topology'', North-Holland. {{isbn|0-444-82432-4}}</ref> Some examples of topics in geometric topology are [[Orientable manifold|orientability]], [[handle decomposition]]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,<ref>{{cite journal | last1 = Johnstone | first1 = Peter T. | author-link = Peter Johnstone (mathematician) | year = 1983 | title = The point of pointless topology | url = http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183550014 | journal = Bulletin of the American Mathematical Society | volume = 8 | issue = 1 | pages = 41–53 | doi = 10.1090/s0273-0979-1983-15080-2 | doi-access = free | archive-date = 27 February 2021 | access-date = 13 January 2014 | archive-url = https://web.archive.org/web/20210227011239/https://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams%2F1183550014 | url-status = live }}</ref> 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.<ref>{{Cite book | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Grothendieck topologies | publisher=Harvard University, Dept. of Mathematics | year=1962 | zbl=0208.48701 | location=Cambridge, MA }}</ref>

==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]].<ref>{{Cite book|first=Colin |last= Adams|author-link=Colin Adams (mathematician)|title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |publisher=American Mathematical Society|year=2004|isbn=978-0-8218-3678-1}}</ref>

===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 complex]]es, indexed by a proximity parameter.
# Analyse these topological complexes via algebraic topology – specifically, via the theory of [[persistent homology]].<ref name=carlsson2009>{{cite journal|url=http://www.ams.org/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf|title=Topology and data|author=Gunnar Carlsson|journal=Bulletin of the American Mathematical Society|series=New Series|volume=46|issue=2|date=April 2009|pages=255–308|doi=10.1090/S0273-0979-09-01249-X|doi-access=free|archive-date=20 March 2009|access-date=30 December 2013|archive-url=https://web.archive.org/web/20090320055948/http://www.ams.org/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf|url-status=live}}</ref>
# Encode the persistent homology of a data set in the form of a parameterized version of a [[Betti number]], which is called a barcode.<ref name=carlsson2009/>

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 algebra]]s over open sets, which are characterized as [[semidecidable]] (equivalently, finitely observable) properties.<ref>{{cite book|title=Topology via Logic|last=Vickers|first=Steve|series=Cambridge Tracts in Theoretical Computer Science|publisher=Cambridge University Press|year=1996|isbn=978-0-521-57651-2}}</ref>

=== Physics ===
Topology is relevant to physics in areas such as [[condensed matter physics]],<ref>{{cite web|title=The Nobel Prize in Physics 2016|url=https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html|publisher=Nobel Foundation|date=4 October 2016|access-date=12 October 2016|archive-date=6 October 2016|archive-url=https://web.archive.org/web/20161006230055/https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html|url-status=live}}</ref> [[quantum field theory]], [[quantum computing]] and [[physical cosmology]].

The topological dependence of mechanical properties in solids is of interest in the 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.<ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|article-number=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> 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.<ref>{{cite journal |last1=Cambou| first1=Anne Dominique |last2=Narayanan | first2=Menon| title=Three-dimensional structure of a sheet crumpled into a ball.| journal= Proceedings of the National Academy of Sciences of the United States of America |year=2011 | volume=108 |issue=36 |pages=14741–14745 | doi=10.1073/pnas.1019192108| pmid=21873249 |arxiv=1203.5826|bibcode=2011PNAS..10814741C|pmc=3169141| doi-access=free }}</ref> 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 invariant]]s. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, [[knot theory]], the theory of [[four-manifold]]s in algebraic topology, and 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 Medal]]s for work related to topological field theory.

The topological classification of [[Calabi–Yau manifold]]s has important implications in [[string theory]], as different manifolds can sustain different kinds of strings.<ref>Yau, S. & Nadis, S.; ''The Shape of Inner Space'', Basic Books, 2010.</ref>

In [[topological quantum computer]]s, the qubits are stored in [[topological properties]], that are by definition invariant with respect to [[homotopies]].<ref>{{cite journal |last=Kitaev |first=Alexei |date=9 July 1997 |title=Fault-tolerant quantum computation by anyons |journal= Annals of Physics|volume=303 |issue=1 |pages=2–30 |doi=10.1016/S0003-4916(02)00018-0 |arxiv=quant-ph/9707021v1|bibcode=2003AnPhy.303....2K |s2cid=11199664 }}</ref>

In cosmology, topology can be used to describe the overall [[shape of the universe]].<ref>''The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds'' 2nd ed (Marcel Dekker, 1985, {{isbn|0-8247-7437-X}})</ref> This area of research is commonly known as [[spacetime topology]].

In [[Condensed matter physics|condensed matter]], a relevant application to topological physics comes from the possibility of obtaining a 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.<ref>{{Cite journal |last1=Haldane |first1=F. D. M. |last2=Raghu |first2=S. |date=2008-01-10 |title=Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry |url=https://link.aps.org/doi/10.1103/PhysRevLett.100.013904 |journal=Physical Review Letters |language=en |volume=100 |issue=1 |article-number=013904 |doi=10.1103/PhysRevLett.100.013904 |pmid=18232766 |arxiv=cond-mat/0503588 |bibcode=2008PhRvL.100a3904H |s2cid=44745453 |issn=0031-9007}}</ref> [[David Thouless]], [[Duncan Haldane]], and [[Michael Kosterlitz]] were awarded the 2016 [[Nobel Prize|Nobel Prize in Physics]] for their work on [[Topological order]]s.<ref>{{cite journal |last1=Gibney |first1=Elizabeth |last2=Castelvecchi |first2=Davide |title=Physics of 2D exotic matter wins Nobel |journal=Nature |date=1 October 2016 |volume=538 |issue=7623 |page=18 |doi=10.1038/nature.2016.20722 |url=https://www.nature.com/articles/nature.2016.20722 |access-date=29 June 2025 |language=en |issn=1476-4687 |archive-date=19 January 2025 |archive-url=https://web.archive.org/web/20250119000110/https://www.nature.com/articles/nature.2016.20722 |url-status=live }}</ref>

===Robotics===
The possible positions of a [[robot]] can be described by a [[manifold]] called [[Configuration space (physics)|configuration space]].<ref>John J. Craig, ''Introduction to Robotics: Mechanics and Control'', 3rd Ed. Prentice-Hall, 2004</ref> In the area of [[motion planning]], one finds paths between two points in configuration space. These paths represent a motion of the robot's [[joint]]s and other parts into the desired pose.<ref>{{cite book|last=Farber|first=Michael|title=Invitation to Topological Robotics|publisher=European Mathematical Society|date=2008|isbn=978-3-03719-054-8}}</ref>

===Games and puzzles===
[[Disentanglement puzzle]]s are based on topological aspects of the puzzle's shapes and components.<ref>{{cite journal |jstor=27642974 |doi=10.2307/27642974 |title= Disentangling Topological Puzzles by Using Knot Theory |first=Mathew |last=Horak |journal=Mathematics Magazine |year=2006 |volume=79 |issue=5 |pages=368–375}}</ref><ref>[http://sma.epfl.ch/~ojangure/Notes.pdf http://sma.epfl.ch/Notes.pdf] {{Webarchive|url=https://web.archive.org/web/20221101044934/https://sma.epfl.ch/~ojangure/Notes.pdf |date=1 November 2022 }} A Topological Puzzle, Inta Bertuccioni, December 2003.</ref><ref>[https://www.futilitycloset.com/2012/06/23/the-figure-8-puzzle/ https://www.futilitycloset.com/the-figure-8-puzzle] {{Webarchive|url=https://web.archive.org/web/20170525030117/http://www.futilitycloset.com/2012/06/23/the-figure-8-puzzle/ |date=25 May 2017 }} The Figure Eight Puzzle, Science and Math, June 2012.</ref>

===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 that surrounds each piece and traverses each edge only once. This process is an application of the [[Eulerian path]].<ref>{{cite book|last=Eckman|first=Edie|title=Connect the shapes crochet motifs: creative techniques for joining motifs of all shapes|date=2012|publisher=Storey Publishing|isbn=978-1-60342-973-3}}</ref>

== Resources and research ==

=== Major journals ===

* ''[[Geometry & Topology]]-'' a [[Scientific journal|mathematic research journal]] focused on geometry and topology, and their applications, published by [[Mathematical Sciences Publishers]]. 
* ''[[Journal of Topology]]-'' a [[scientific journal]] which publishes papers of high quality and significance in topology, [[geometry]], and adjacent areas of mathematics.

=== Major books ===

* Munkres, James R. (2000). ''Topology'' (2nd ed.). Upper Saddle River, NJ: Prentice Hall. {{ISBN|978-0-13-181629-9}}
* Willard, Stephen (2016). ''General topology''. Dover books on mathematics. Mineola, N.Y: Dover publications. {{ISBN|978-0-486-43479-7}}
* Armstrong, M. A. (1983). ''Basic topology''. Undergraduate texts in mathematics. New York: Springer-Verlag. {{ISBN|978-0-387-90839-7}}
* John Kelley "General Topology" Springer, 1979.

==See also==
{{Portal|Mathematics}}
{{cols|colwidth=21em}}
* [[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]]
{{colend}}

==References==
===Citations===
{{reflist}}

===Bibliography===
* {{citation|last=Aleksandrov|first=P.S.|chapter=Chapter XVIII Topology|editor1-first=A.D.|editor1-last=Aleksandrov|editor2-first=A.N.|editor2-last=Kolmogorov|editor3-first=M.A.|editor3-last=Lavrent'ev|title=Mathematics / Its Content, Methods and Meaning|edition=2nd|publisher=The M.I.T. Press|year=1969|orig-date=1956}}
* {{citation|last=Croom|first=Fred H.|title=Principles of Topology|year=1989|publisher=Saunders College Publishing|isbn=978-0-03-029804-2}}
* {{citation|last=Richeson|first= D.|author-link= David Richeson |title=Euler's Gem: The Polyhedron Formula and the Birth of Topology|publisher=Princeton University Press|year=2008|title-link= Euler's Gem}}

==Further reading==
* [[Ryszard Engelking]], ''General Topology'', Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, {{isbn|3-88538-006-4}}.
* [[Nicolas Bourbaki|Bourbaki]]; ''Elements of Mathematics: General Topology'', Addison–Wesley (1966).
* {{cite book| last = Breitenberger| first = E.| year = 2006| chapter = Johann Benedict Listing| title = History of Topology| editor-last = James| editor-first = I.M.| publisher = North Holland| isbn = 978-0-444-82375-5}}
* {{Kelley 1975}}
* {{cite book| last = Brown| first = Ronald| author-link = Ronald Brown (mathematician)| year = 2006| title = Topology and Groupoids| url= http://pages.bangor.ac.uk/~mas010/topgpds.html| publisher = Booksurge| isbn = 978-1-4196-2722-4}} (Provides a well-motivated, geometric account of general topology, and shows the use of groupoids in discussing [[van Kampen's theorem]], [[covering space]]s, and [[orbit space]]s.)
* [[Wacław Sierpiński]], ''General Topology'', Dover Publications, 2000, {{isbn|0-486-41148-6}}
* {{cite book| last = Pickover| first = Clifford A.| author-link = Clifford A. Pickover| year = 2006| title = The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology| publisher = Thunder's Mouth Press| isbn = 978-1-56025-826-1| url-access = registration| url = https://archive.org/details/mbiusstripdrau00pick}} (Provides a popular introduction to topology and geometry)
* {{citation|first=Michael C.|last=Gemignani|title=Elementary Topology|edition=2nd|year=1990|orig-date=1967|publisher=Dover Publications Inc.|isbn=978-0-486-66522-1}}

==External links==
{{Commons|Topology}}
{{Wikiquote}}
{{Wikibooks}}
* {{Springer |title=Topology, general |id=p/t093200}}
* [http://www.pdmi.ras.ru/~olegviro/topoman/index.html Elementary Topology: A First Course] Viro, Ivanov, Netsvetaev, Kharlamov.
* [http://www.geom.uiuc.edu/zoo/ The Topological Zoo] at [[The Geometry Center]].
* [http://at.yorku.ca/index.html Topology Atlas] {{Webarchive|url=https://web.archive.org/web/20240428164301/http://at.yorku.ca/index.html |date=28 April 2024 }}
* [http://at.yorku.ca/i/a/a/b/23.htm Topology Course Lecture Notes] {{Webarchive|url=https://web.archive.org/web/20110805165214/http://at.yorku.ca/i/a/a/b/23.htm |date=5 August 2011 }} Aisling McCluskey and Brian McMaster, Topology Atlas.
* [https://web.archive.org/web/20090713073050/http://www.ornl.gov/sci/ortep/topology/defs.txt Topology Glossary]
* [https://link.springer.com/chapter/10.1007/978-1-4612-2972-8_1 Moscow 1935: Topology moving towards America], a historical essay by [[Hassler Whitney]].

{{Topology |expanded}}
{{Areas of mathematics |collapsed}}
{{Authority control}}

[[Category:Topology| ]]
[[Category:Mathematical structures]]