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Topological definition

It is a branch of mathematics developed in modern times to study continuity phenomena. Chinese names originated from Greek τ ο π ο λ ο γ? Transliteration of α. Topology, originally meaning landform, was introduced by scientists in the middle of19th century. At that time, I mainly studied some geometric problems arising from the needs of mathematical analysis. So far, topology mainly studies the invariant properties and invariants of topological space under topological transformation.

For example, in the usual plane geometry, if one figure on the plane moves to another figure, if they are completely coincident, then the two figures are said to be conformal. However, the graph studied in topology changes in motion, regardless of its size or shape. In topology, there is no element that cannot be bent, and the size and shape of each figure can be changed. For example, when Euler solved the problem of the Seven Bridges in Konigsberg, he did not consider its size and shape, but only the number of points and lines. These are the starting points of topological thinking.

Simply put, topology is to study how tangible objects keep their properties unchanged under continuous transformation.

Edit the topological properties of this paragraph.

What are the topological properties? Firstly, we introduce topological equivalence, which is an easy-to-understand topological property.

Topology does not discuss the concept of congruence between two graphs, but the concept of topological equivalence. For example, circles, squares and triangles are all equivalent graphs under topological transformation, although their shapes and sizes are different. In other words, from a topological point of view, they are exactly the same.

Select some points on a sphere and connect them with disjoint lines, so that the sphere is divided into many blocks by these lines. Under topological transformation, the number of points, lines and blocks is still the same as the original number, which is topological equivalence. Generally speaking, for a closed surface with arbitrary shape, as long as the surface is not torn or cut, its transformation is topological change, and topological equivalence exists.

It should be pointed out that torus does not have this property. For example, if the torus is cut as shown on the left, it will not be divided into many pieces, but just become a curved bucket. In this case, we say that a sphere cannot be topologically torus. So sphere and torus are topologically different surfaces.

The combination relationship and order relationship between points and lines on a straight line remain unchanged under topological transformation, which is a topological property. In topology, the closed properties of curves and surfaces are also topological properties.

The plane and surface we usually talk about usually have two sides, just like a piece of paper has two sides. But the German mathematician Mobius (1790 ~ 1868) discovered Mobius surface in 1858. This surface cannot be painted with different colors on both sides.

There are many invariants and invariants of topological transformation, which are not introduced here.

Edit the topology development of this section.

After the establishment of topology, it has also developed rapidly due to the development needs of other mathematical disciplines. Especially after Riemann founded Riemann geometry, he took the concept of topology as the basis of analytic function theory, which further promoted the progress of topology.

Since the twentieth century, set theory has been introduced into topology, which has opened up a new look for topology. Topology becomes the corresponding concept of arbitrary point set. Some problems that need to be accurately described in topology can be discussed by sets.

Because a large number of natural phenomena are continuous, topology has the possibility of extensive contact with various practical things. Through the study of topology, we can make clear the set structure of space and grasp the functional relationship between spaces. Since 1930s, mathematicians have made more in-depth research on topology and put forward many new concepts. Such as uniform structure, abstract distance, approximate space, etc. There is a branch of mathematics called differential geometry, which uses differential tools to study the bending of lines and surfaces near a point, and topology studies the global relationship of surfaces. Therefore, there should be some essential connection between these two disciplines. 1945, China mathematician Chen Shengshen established the connection between algebraic topology and differential geometry, which promoted the development of global geometry.

Until today, topology has been divided into two branches in theory. One branch focuses on analytical methods, called point set topology, or analytical topology. Another branch focuses on algebraic methods, called algebraic topology. Now, these two branches have a unified trend.

Topology is widely used in functional analysis, Lie group theory, differential geometry, differential equations and many other branches of mathematics.

Edit the brief history of this paragraph.

Topology was first called situation analysis, which was translated by G.W. Leibniz in 1679 (translated into Chinese, situation refers to the nature of a graph itself, and potential refers to the relative nature of a graph and its subgraphs. Post-complement (consistency space, paracompactness, etc. ) and the arrangement, knot and embedding of Bourbaki School in the mid-1930s are potential problems). Subsequently, the Polish School and the Soviet School systematically studied the basic properties of topological space (separability, compactness, connectivity, etc.). ). l Euler solved the seven-bridge problem in 1736 and published the polyhedron formula in 1750; C.F. Gauss 1833 defines the number of circles of two closed curves in space by line integral in electrodynamics. The word topology (Chinese transliteration) was put forward by J.B. Listing (1847), which comes from Greek (location, situation) and (knowledge). This is the embryonic stage.

Starting from 185 1, B. Riemann put forward the geometric concept of Riemann surface in the study of complex variable functions, and stressed that in order to study functions and integrals, it is necessary to study situation analysis. From then on, a systematic study of topology began. Under the influence of point set theory, Riemann himself solved the homeomorphism classification problem of orientable closed surfaces. Such as gathering point (limit point), open set, closed set, density, connectivity and so on. In the study of geometry, Riemann clearly put forward the concept of n-dimensional manifold (1854). Many topological concepts are obtained,

The founder of combinatorial topology is h poincare. In his analysis and mechanical work, especially in the study of univalence of complex functions and curves determined by differential equations, he leads to topological problems, but his methods are sometimes not rigorous enough, and his main interest is n-dimensional manifolds. During 1895 ~ 1904, he established the basic method of studying manifolds by subdivision. He introduced many invariants: basic group, homology, Betty number and torsion coefficient, and put forward specific calculation methods. He introduced many invariants: fundamental group, homology, Betty number and torsion coefficient. He discussed the topological classification of three-dimensional manifolds and put forward the famous Poincare conjecture. The rich thoughts he left have far-reaching influence, but his methods are sometimes not rigorous enough and rely too much on geometric intuition. Especially when studying complex functions and single values of curves determined by differential equations,

Another source of topology is the rigor of analysis. It is in his book Analysis and Mechanics that the strict definition of real numbers pushes G. Cantor to systematically study point sets in Euclidean space from 1873, and obtains many topological concepts, such as gathering points (extreme points), open sets, closed sets, density, connectivity and so on. Under the influence of point set theory, the concept of universal function (that is, function) appeared in the analysis. Taking the function set as a geometric object, its limit was discussed. This eventually leads to the concept of abstract space. In this way, B. Riemann put forward the geometric concept of Riemann surface in the study of complex variable functions. At the turn of the 20th century, two research directions, combinatorial topology and point set topology, were formed. This is the embryonic stage.

General topology was first studied by Fréchet, who introduced the concept of metric space in 1906. F Hausdorf defined a relatively general topological space with an open neighborhood in the Outline of Set Theory (19 14), which marked the emergence of general topology which used axiomatic method to study continuity. L Euler solved the seven-bridge problem in 1736, and then Polish school and Soviet school systematically studied the basic properties of topological space (separability, compactness, connectivity, etc.). ). Post-complement (consistency space, paracompactness, etc. ) and the arrangement of the Bourbaki School Since the mid-1930s, general topology has matured and become the same basis for mathematical research after the Second World War. From the influence of its methods and results on mathematics, the theory of compact topological space and complete metric space is the most important. The compactness and quantification problems are also deeply studied. The recent development of axiomatic general topology can be found in General Topology.

For example, the study of point set in Euclidean space has always been an important part of topology, which has developed to the intersection of general topology and algebraic topology, and can also be regarded as a part of geometric topology. Since 1950s, the work of American school, represented by R.H. Bing, has deepened the understanding of manifolds and raised the question of whether two given mappings are homotopy, which has played a role in proving the four-dimensional Poincare conjecture. The study of dimension and continuum caused by peano curve is also regarded as a branch of general topology.

Algebraic topology L.E.J Brouwer proposed a method of approximating continuous mapping with simple mapping during1910 ~1912. Many important geometric phenomena are used to prove that Euclidean space with different dimensions has different embryos, so it is different embryos. The mapping degree between manifolds of the same dimension is introduced to study homotopy classification, and the fixed point theory is established. He made combinatorial topology reach its due level in terms of precise concept and rigorous demonstration, while Euler number υ-e+? & gt yes). Become a compelling theme. Then, J.W. Alexander proved the topological invariance of Betty number and torsion coefficient in 19 15. E.g., connectivity, compactness),

With the rise of abstract algebra, around 1925, A.E. Nott proposed to establish combinatorial topology on the basis of group theory. Under her influence, H. hopf defined the homology group in 1928. Since then, combinatorial topology has gradually evolved into algebraic topology that uses abstract algebra to study topological problems. For example, dimensions and Euler numbers, S. Allen Berg and N. E. Stranrod summed up the theory of homology at that time in an axiomatic way in 1945, and later wrote "Fundamentals of Algebraic Topology" (1952), which greatly promoted the spread, application and further development of algebraic topology. They summarized the basic spirit of algebraic topology as: transforming topological problems into algebraic problems and solving them through calculation. Homology groups and cohomology rings introduced in 1930s are the transition from topology to algebra (see homology theory). Until today, triangles and circles are homeomorphic; The invariants provided by homology theory (including cohomology) are still the easiest to calculate in topology, so they are also the most commonly used. There is no need to make a distinction.

Homotopy ethics studies the homotopy classification of space and mapping. W. leonid hurwicz introduced the N-dimensional homotopy group of topological space from 1935 to 1936, whose elements are homotopy classes of the mapping from the N-dimensional sphere to this space, and? Inverse mapping with it? -1:B→A are all continuous, and one-dimensional homotopy groups are just basic groups. Homotopy group provides another transition from topology to algebra, which exactly means homeomorphism. Its geometric meaning is more obvious than homology group. The continuous deformation of the geometry mentioned above is extremely difficult to calculate. The calculation of homotopy groups, especially spherical homotopy groups, promotes the development of topology and produces colorful theories and methods. 1950, J.P. Searle studied the homology theory of fiber bundles by using the spectral sequence algebra tool developed by J.Leray, and the simplest example is Euclidean space. The calculation of homotopy group has made a breakthrough, which opens the way for the rapid development of topology.

At the end of 1950s, under the influence of algebraic geometry and differential topology, K theory came into being, which solved a series of topological problems about manifolds, and several generalized homology theories appeared. It is a transition from topology to algebra, and algebra is a generalized geometric figure. Although the geometric meaning is different, for example, all possible states of a system in physics constitute the so-called state space, but the algebraic properties are very similar to homology or cohomology, which is a powerful weapon of algebraic topology. Theoretically, it is obvious that homology theory (ordinary and generalized) is essentially a part of homotopy theory.

From differential topology to geometric topology, differential topology is a topology that studies differential manifolds and differential mappings. These properties are independent of length and angle. Lagrange and riemann sum Poincare have studied differential manifolds. With the progress of algebraic topology and differential geometry, these examples reveal that geometric figures have some properties that cannot be studied by traditional geometric methods. It reappeared in the 1930s. H Whitney 1935 gives a general definition of differential manifold, and proves that it can always be embedded in high-dimensional Euclidean space as a smooth submanifold. In order to study the vector field on differential manifold, he also put forward the concept of fiber bundle, which linked many geometric problems with cohomology (indicator class) and homotopy problems.

1953, R. Thom's synergetic theory (see differential topology) created a situation that differential topology and algebraic topology kept pace, and many difficult differential topology problems were solved by transforming them into algebraic topology problems, which also stimulated the further development of algebraic topology. The number of rotations of the vector from the driving point to its image point. 1956, J.W. Milnor discovered that there are unusual differential structures on the seven-dimensional sphere besides the usual differential structures. Each fixed point also has an "exponent", and then a manifold that cannot be endowed with any differential structure is reconstructed, which shows that there is a great difference between topological manifold, differential manifold and piecewise linear manifold between them, and differential topology is recognized as an independent branch of topology from now on. 1960, S. Smale proved the Poincare conjecture of differential manifolds with more than five dimensions. J.W. Milnor and others have developed the basic method of dealing with differential manifolds-planing technology, which makes the classification of manifolds with more than five dimensions tend to algebra gradually.

In recent years, manifold research has made a lot of progress, such as the discipline of geometry and geometric methods. Prominent places such as the relationship between the above three types of manifolds and the classification of three-dimensional and four-dimensional manifolds. Major achievements in the early 1980s include the proof of four-dimensional Poincare conjecture and the discovery of unusual differential structures in four-dimensional Euclidean space. This kind of research is usually called geometric topology to emphasize its geometric color, but a vector field without singularity can be created on the torus. Different from the homoethics with strong algebraic flavor.

The relationship between topology and other disciplines: continuity and discreteness. This contradiction generally exists in natural and social phenomena, and mathematics can be roughly divided into two categories: continuity and discreteness. Topology is of fundamental significance to continuous mathematics and greatly promotes discrete mathematics. For example, the basic content of topology has become the common sense of modern mathematicians. The importance of topology lies in its interaction with other branches of mathematics and other disciplines.

Topology is closely related to differential geometry,/img/Y2Hl2JHawtl2RizevdjevzmlndxjL3N4DHVCDA 0LMPWZW = =. JPG Target = " _ blank " > & lt； img src =/img/y2hhl 2 jhawtl 2 rizevdjevzmlndxjl3n 4 dhvcda 0 lmpwzw =。 Jpg vector field problem "> vector field problem considers the continuous tangent vector field on a smooth surface and studies the properties of manifolds at different levels. It depends on whether one of these two graphs is not included. In order to study geodesics on Riemannian manifolds and whether a network can be embedded in a plane, H.M. Morse established the nondegenerate critical point theory in the 1920s, which linked the exponent of critical point of smooth function on manifolds with Betty number of manifolds and developed it into a large-scale variational method. Morse theory was later applied to topology, which proved the Bote periodicity of homotopy groups of typical groups (this is the cornerstone of K theory) and inspired the complementary technology to deal with differential manifolds. Differential manifold, fiber bundle, characteristic class. Adam's global differential geometry provides a suitable theoretical framework, from which he has gained a strong impetus and rich topics. G. piano created such a "curve" in 1890, and Chen Shengshen introduced the "demonstration class" in the 1940s, which not only had a far-reaching impact on differential geometry, but also the trajectory described by a moving point with continuously changing parameters (time) was a curve. It is also important for topology. The simple concept is that points move into a line. Fiber bundle theory and connection theory together provide a ready-made mathematical framework for Young-Mills gauge field theory in theoretical physics (see Young-Mills theory). Dimension problem

Topology has greatly promoted the development of modern analytical science. With the development of science and technology, it is necessary to study various nonlinear phenomena, and the analysis depends more on topology. To ask whether a knot can be untied (that is, whether it can be transformed into a flat circle), J. Leray and J. P. Sauder extended L. E. J. Brouwer's fixed point theorem and mapping degree theory to Banach spaces in the 1930s, forming a topological degree theory. The latter and the critical point theory mentioned above, kink problem ">; The self-disjoint closed curve in the knot problem space has become a standard tool for studying nonlinear partial differential equations. Therefore, this chromatic number is also an invariant property of surfaces under continuous deformation. The progress of differential topology promotes the development of analysis to manifold analysis (also called large-scale analysis). Under the influence of Thom, it was distorted at will, and the structural stability theory and singularity theory of differential mapping developed into important branches. The theory of differential dynamic system founded by S. Smale in the early 1960s needs seven colors. It is the theory of ordinary differential equations on manifolds. In the early 1960s, M.F. Atia and others established the theory of elliptic operators on differential manifolds. The famous Atiya-Singer index theorem relates the analytic index of operators to the characteristic classes of manifolds, which is an example of the combination of analysis and topology. The operator algebra of modern functional analysis is closely related to K theory, exponential theory and leaf structure. Layer theory from algebraic topology has become a basic tool in the theory of multiple complex variables.

The need of topology has greatly stimulated the development of abstract algebra, forming two new branches of algebra: homology algebra and algebraic K theory. The four-color problem draws a map on a plane or a sphere. Algebraic geometry has completely changed since the 1950s. The Euler number υ-e+ after the surface is transformed into a polyhedron? It plays a key role (see/baike/%ca% FD% d1%a7 _% b1%D5% C7% fa% C3% E6% b5% C4% B7% D6% c0% E0.html target = _ blank > classification of closed surfaces). Thom's synergetic theory directly promoted the production of Riemann-Roche theorem of algebraic clusters, and then promoted the production of topological K theory. Modern algebraic geometry has completely used homology language. How many different types of closed surfaces are there under continuous deformation? On this basis, algebraic number theory and algebraic groups have also made many important achievements, such as the proof of Weii conjecture and Mo Deer conjecture on the number estimation of integer solutions of indefinite equations (see algebraic number theory).

The concepts of category and functor are formed when summarizing algebraic topological methodology. Category theory has penetrated into the branches of mathematical foundation and algebraic geometry (see category); It also has an impact on the topology itself. The popular saying is that there are holes in the picture frame. For example, the concept of topology greatly broadens the classical concept of topological space. There are more essential differences between convex and frame shapes than long and short curves.

In economics, this shows that J von Neumann first proved the existence of equilibrium with the fixed point theorem. In modern mathematical economics, the existence, nature and calculation of economic mathematical models are inseparable from tools such as algebraic topology, differential topology and large-scale analysis. Topology also has important applications in system theory, game theory, planning theory and network theory.

Thom founded catastrophe theory based on the singularity theory of differential mapping in differential topology, which provided various mathematical models for the transformation from quantitative to qualitative. In physics, chemistry, biology, linguistics and other aspects, there have been many applications of "Euler polyhedron formula and classification of surfaces" >: Eurasian discovery of Euler polyhedron formula and surfaces,

The concepts and methods of topology are not only indirectly influenced by various branches of mathematics, but also have direct applications to physics (such as the classification of liquid crystal structural defects), chemistry (such as the topological configuration of molecules) and biology (such as DNA encapsulation and topoisomerase).

The marginal research between topology, mathematics and science is in the ascendant.