s, which have only one surface and one edge, are a kind of object studied in topology.]]
, topology (from the Greek
words , and ) is concerned with the properties of a geometric object
that are preserved under continuous deformations
, such as stretching
, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
A topological space
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
s, and, more generally, metric space
s are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphism
s and homotopies
. A property that is invariant under such deformations is a topological property
. Basic examples of topological properties are: the dimension
, which allows distinguishing between a line
and a surface
, which allows distinguishing between a line and a circle; connectedness
, which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Leibniz
, who in the 17th century envisioned the and . Leonhard Euler
's Seven Bridges of Königsberg
problem and polyhedron formula
are arguably the field's first theorems. The term ''topology'' was introduced by Johann Benedict Listing
in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.
, the simplest non-trivial knot
The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad
) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg
problem led to the branch of mathematics known as graph theory
Similarly, the hairy ball theorem
of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick
." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field
on the sphere. As with the ''Bridges of Königsberg'', the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes.
To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.
Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
Homeomorphism can be considered the most basic topological equivalence
. Another is homotopy equivalence
. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.
An introductory exercise
is to classify the uppercase letters of the English alphabet
according to homeomorphism and homotopy equivalence. The result depends on the font used, and on whether the strokes making up the letters have some thickness or are ideal curves with no thickness. The figures here use the sans-serif Myriad
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
, is different on the supposedly differing classes.
Letter topology has practical relevance in stencil typography
. For instance, Braggadocio
font stencils are made of one connected piece of material.
thumb|right|240px|The Seven Bridges of Königsberg
was a problem solved by Euler.
Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.
Among these are certain questions in geometry investigated by Leonhard Euler
. His 1736 paper on the Seven Bridges of Königsberg
is regarded as one of the first practical applications of topology.
On 14 November 1750, Euler wrote to a friend that he had realized the importance of the ''edges'' of a polyhedron
. This led to his polyhedron formula
, (where , , and respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology.
Further contributions were made by Augustin-Louis Cauchy
, Ludwig Schläfli
, Johann Benedict Listing
, Bernhard Riemann
and Enrico Betti
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
. In 1895, he published his ground-breaking paper on ''Analysis Situs
'', which introduced the concepts now known as homotopy
, which are now considered part of algebraic topology
Unifying the work on function spaces of Georg Cantor
, Vito Volterra
, Cesare Arzelà
, Jacques Hadamard
, Giulio Ascoli
and others, Maurice Fréchet
introduced the metric space
in 1906. A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff
coined the term "topological space" and gave the definition for what is now called a Hausdorff space
. Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space
as part of his study of Fourier series
. For further developments, see point-set topology
and algebraic topology.
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
, the complex plane
, and the Cantor set
can be thought of as the same set with different topologies.
Formally, let be a set and let be a family
of subsets of . Then is called a topology on if:
# Both the empty set and are elements of .
# Any union of elements of is an element of .
# Any intersection of finitely many elements of is an element of .
If is a topology on , then the pair is called a topological space. The notation may be used to denote a set endowed with the particular topology . By definition, every topology is a -system
The members of are called ''open sets'' in . A subset of is said to be closed if its complement is in (that is, its complement is open). A subset of may be open, closed, both (a clopen set
), or neither. The empty set and itself are always both closed and open. An open subset of which contains a point is called a neighborhood
Continuous functions and homeomorphisms
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 one-to-one
, 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.
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
that is homeomorphic
to the Euclidean space of dimension . Lines
s, but not figure eights
, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces
, although not all surfaces
are manifolds. Examples include the plane
, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle
and real projective plane
, which cannot (that is, all their realizations are surfaces that are not manifolds).
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
s, which are sets equipped with a topology
, that is, a family of subset
s, called ''open sets'', which is closed
under finite intersection
s and (finite or infinite) union
s. The fundamental concepts of topology, such as ''continuity
'', and ''connectedness
'', can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words ''nearby'', ''arbitrarily small'', and ''far apart'' can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected.
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
, the complex plane
, real and complex vector space
s and Euclidean space
s. Having a metric simplifies many proofs.
Algebraic topology is a branch of mathematics that uses tools from algebra
to study topological spaces. The basic goal is to find algebraic invariants that classify
topological spaces up to
homeomorphism, though usually most classify up to homotopy equivalence.
The most important of these invariants are homotopy group
s, homology, and cohomology
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group
is again a free group.
Differential topology is the field dealing with differentiable function
s on differentiable manifold
s. 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 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 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. Some examples of topics in geometric topology are orientability
, handle decomposition
s, local flatness
, crumpling and the planar and higher-dimensional 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 (Riemann
surfaces are complex curves) – by the uniformization theorem every conformal class
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. In pointless topology
one considers instead the lattice
of open sets as the basic notion of the theory, while Grothendieck topologies
are structures defined on arbitrary categories
that allow the definition of sheaves
on those categories, and with that the definition of general cohomology theories.
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
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.
Topological data analysis
uses techniques from algebraic topology to determine the large scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal
). The main method used by topological data analysis is to:
# Replace a set of data points with a family of simplicial complex
es, indexed by a proximity parameter.
# Analyse these topological complexes via algebraic topology – specifically, via the theory of persistent homology
# Encode the persistent homology of a data set in the form of a parameterized version of a Betti number
, which is called a barcode.
Several branches of programming language semantics, such as domain theory, are formalized using topology. In this context, Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.
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 crumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space. Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics.
A topological quantum field theory (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. Donaldson, Jones, Witten, and 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.
The possible positions of a robot can be described by a manifold called configuration space. In the area of motion planning, one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose.
Games and puzzles
Tanglement puzzles are based on topological aspects of the puzzle's shapes and components.
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.
* 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
* Publications in topology
* Topology glossary
* Topological geometry
* Topological order
* Ryszard Engelking, ''General Topology'', Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, .
* Bourbaki; ''Elements of Mathematics: General Topology'', Addison–Wesley (1966).
* (Provides a well motivated, geometric account of general topology, and shows the use of groupoids in discussing van Kampen's theorem, covering spaces, and orbit spaces.)
* Wacław Sierpiński, ''General Topology'', Dover Publications, 2000,
* (Provides a popular introduction to topology and geometry)
Elementary Topology: A First Course
Viro, Ivanov, Netsvetaev, Kharlamov.
The Topological Zoo
at The Geometry Center.
Aisling McCluskey and Brian McMaster, Topology Atlas.
Moscow 1935: Topology moving towards America
a historical essay by Hassler Whitney.