Topology (from theGreek wordsτόπος,'place, location', andλόγος,'study') is the branch ofmathematics concerned with the properties of ageometric object that are preserved undercontinuousdeformations, such asstretching,twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
Atopological space is aset endowed with a structure, called atopology, which allows defining continuous deformation of subspaces, and, more generally, all kinds ofcontinuity.Euclidean spaces, and, more generally,metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology arehomeomorphisms andhomotopies. A property that is invariant under such deformations is atopological property. The following are basic examples of topological properties: thedimension, which allows distinguishing between aline and asurface;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 toGottfried Wilhelm Leibniz, who in the 17th century envisioned thegeometria situs andanalysis situs.Leonhard Euler'sSeven Bridges of Königsberg problem andpolyhedron formula are arguably the field's first theorems. The termtopology was introduced byJohann 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.
Möbius strips, 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 (nowKaliningrad) that would cross each of its seven bridges exactly once.[1] 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. ThisSeven Bridges of Königsberg problem led to the branch of mathematics known asgraph theory.[2]
Similarly, thehairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating acowlick."[3] 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 continuoustangent vector field on the sphere. As with theBridges 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 ofhomeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridgeshomeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.
A continuous transformation can turn a coffee mug into a donut. Ceramic model by Keenan Crane andHenry 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.[4] A pliabletorus (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.[5]
Homeomorphism can be considered the most basictopological equivalence. Another ishomotopy 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.
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.[6] Among these are certain questions in geometry investigated byLeonhard Euler. His 1736 paper on theSeven Bridges of Königsberg is regarded as one of the first practical applications of topology.[6] On 14 November 1750, Euler wrote to a friend that he had realized the importance of theedges of apolyhedron. This led to hispolyhedron formula,V −E +F = 2 (whereV,E, andF 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.[7]
Further contributions were made byAugustin-Louis Cauchy,Ludwig Schläfli,Johann Benedict Listing,Bernhard Riemann andEnrico Betti.[8] Listing introduced the term "Topologie" inVorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print.[9] The English form "topology" was used in 1883 in Listing's obituary in the journalNature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated".[10]
Their work was corrected, consolidated and greatly extended byHenri Poincaré. In 1895, he published his ground-breaking paper onAnalysis Situs, which introduced the concepts now known ashomotopy andhomology, which are now considered part ofalgebraic topology.[8]
Topological characteristics of closed 2-manifolds[8]
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 ofGeorg Cantor,Vito Volterra,Cesare Arzelà,Jacques Hadamard,Giulio Ascoli and others,Maurice Fréchet introduced themetric space in 1906.[11] 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 aHausdorff space.[12] Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 byKazimierz Kuratowski.[13]
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 inEuclidean space as part of his study ofFourier series. For further developments, seepoint-set topology and algebraic topology.
The 2022Abel Prize was awarded toDennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".[14]
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, thereal line, thecomplex plane, and theCantor set can be thought of as the same set with different topologies.
Formally, letX be a set and letτ be afamily of subsets ofX. Thenτ is called a topology onX if:[15]
Both the empty set andX are elements ofτ.
Any union of elements ofτ is an element ofτ.
Any intersection of finitely many elements ofτ is an element ofτ.
Ifτ is a topology onX, then the pair(X,τ) is called a topological space. The notationXτ may be used to denote a setX endowed with the particular topologyτ. By definition, every topology is aπ-system.
The members ofτ are calledopen sets inX. A subset ofX is said to be closed if its complement is inτ (that is, its complement is open). A subset ofX may be open, closed, both (aclopen set), or neither. The empty set andX itself are always both closed and open. An open subset ofX which contains a pointx is called an openneighborhood ofx.
A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and of a cowplushie into a sphere
Afunction or map from one topological space to another is calledcontinuous if theinverse image of any open set is open. If the function maps thereal numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous incalculus. If a continuous function isone-to-one andonto, 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.
While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. Amanifold is a topological space that resembles Euclidean space near each point. More precisely, each point of ann-dimensional manifold has aneighborhood that ishomeomorphic to the Euclidean space of dimensionn.Lines andcircles, but notfigure eights, are one-dimensional manifolds. Two-dimensional manifolds are also calledsurfaces. Examples include theplane, thesphere, and thetorus, which can all be realized in three dimensions without self-intersection, and theKlein bottle andreal projective plane, which cannot.
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.[16][17] 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 istopological spaces, which are sets equipped with atopology, that is, a family ofsubsets, calledopen sets, which isclosed under finiteintersections and (finite or infinite)unions. The fundamental concepts of topology, such ascontinuity,compactness, andconnectedness, 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 wordsnearby,arbitrarily small, andfar 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 spaces are an important class of topological spaces where the distance between any two points is defined by a function called ametric. In a metric space, an open set is a union of open disks, where an open disk of radiusr centered atx is the set of all points whose distance tox is less thanr. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of thereal line, thecomplex plane, real and complex normedvector spaces andEuclidean spaces. Having a metric simplifies many proofs.
Algebraic topology is a branch of mathematics that uses tools fromalgebra to study topological spaces.[18] The basic goal is to find algebraic invariants thatclassify topological spacesup to homeomorphism, or more commonly up to homotopy equivalence.
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 afree group is again a free group.
More specifically, differential topology considers the properties and structures that require only asmooth 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 anddeformations that exist in differential topology. For instance, volume andRiemannian 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 is a branch of topology that primarily focuses on low-dimensionalmanifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.[20] Some examples of topics in geometric topology areorientability,handle decompositions,local flatness, crumpling and the planar and higher-dimensionalSchönflies theorem.
Low-dimensional topology is strongly geometric, as reflected in theuniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positivecurvature/spherical, zero curvature/flat, and negative curvature/hyperbolic – and thegeometrization 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 ascomplex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem everyconformal class ofmetrics 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.
Occasionally, one needs to use the tools of topology but a "set of points" is not available. Inpointless topology one considers instead thelattice of open sets as the basic notion of the theory,[21] whileGrothendieck topologies are structures defined on arbitrarycategories that allow the definition ofsheaves on those categories and with that the definition of general cohomology theories.[22]
Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular,circuit topology andknot 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 slowerelectrophoresis.[23]
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 ortoroidal). The main method used by topological data analysis is to:
Replace a set of data points with a family ofsimplicial complexes, indexed by a proximity parameter.
Analyse these topological complexes via algebraic topology – specifically, via the theory ofpersistent homology.[24]
Encode the persistent homology of a data set in the form of a parameterized version of aBetti number, which is called a barcode.[24]
The topological dependence of mechanical properties in solids is of interest in the disciplines ofmechanical engineering andmaterials science. Electrical and mechanical properties depend on the arrangement and network structures ofmolecules and elementary units in materials.[27] Thecompressive strength ofcrumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space.[28] Topology is of further significance inContact mechanics where the dependence of stiffness and friction on thedimensionality of surface structures is the subject of interest with applications in multi-body physics.
The topological classification ofCalabi–Yau manifolds has important implications instring theory, as different manifolds can sustain different kinds of strings.[29]
The possible positions of arobot can be described by amanifold calledconfiguration space.[34] In the area ofmotion planning, one finds paths between two points in configuration space. These paths represent a motion of the robot'sjoints and other parts into the desired pose.[35]
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 theEulerian path.[39]
^Artin, Michael (1962).Grothendieck topologies. Cambridge, MA: Harvard University, Dept. of Mathematics.Zbl0208.48701.
^Adams, Colin (2004).The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society.ISBN978-0-8218-3678-1.
Aleksandrov, P.S. (1969) [1956]. "Chapter XVIII Topology". In Aleksandrov, A.D.; Kolmogorov, A.N.; Lavrent'ev, M.A. (eds.).Mathematics / Its Content, Methods and Meaning (2nd ed.). The M.I.T. Press.
Croom, Fred H. (1989).Principles of Topology. Saunders College Publishing.ISBN978-0-03-029804-2.
