Topology of origin _ tilted and bird birds

Geometric topology is the formation of a nineteenth century mathematics branch, it belongs to the category of the geometry. The topology of some content as early as the 18th century. At that time found some isolated problem, and later the formation of the topology in an important position. In mathematics, on Fort VII bridge Yihong, Polyhedra of Euler's theorem, four color issues are important in the history of the topology. Yihong Fort (now Russia Kaliningrad) is the capital of East Prussia, Puerto Allegre River running through it. In the 18th century in the river has seven bridge, River in the middle of the two islands and the Bank link. People's leisure time often in this top for a walk, a day is suggested: can cannot bridge are only walking again, finally returned to its original location. This issue looks very simple and has a very interesting question attracted us, many people try different kinds of way, but no one has done. It seems to be a clear, the ideal answer is not so easy. In 1736, someone with these problems found then mathematicians Euler Euler after some thoughts on soon with a unique approach gives the answer. Euler and put this issue first simplify, he put the two small islands and the River were seen as a fourth point, the seven bridges as these four points on a line. So the problem is reduced to, can use a pen to draw the graphics. After further analysis, Euler concluded �� not likely each bridge will go again, finally back to its original location. And gives all to draw out the graphic should have conditions. This is the "Harbinger" topology. In the history of topology, there is also a famous and important theorems on Polyhedra and Euler. This theorem: If a convex polyhedron vertices is v, edge is e, surface is f, they always have such a relationship: f + v-e = 2. According to the polyhedron Euler's theorem, it can be concluded that an interesting fact: only five regular polyhedra. They are tetrahedral, are hexahedral, octahedron, dodecahedron, icosahedron. The famous "four color problem" is also relevant to the development of the topology. Four color issues also known as the four color guess, is the world's largest math problems. Four color conjecture of appeal from United Kingdom. In 1852, he graduated from London University Fernando Francis Guthrie come to an institution engaged in map rendering, found an interesting phenomenon: "it appears that each map can be used four color rendering, making a common border States are a different color. " 1872, the United Kingdom at that time the most famous mathematician Kelly officially to the London Mathematical Society asked this question, and a four-color guess became world mathematics community concerns. Many of the world's leading mathematicians have participated in the four colors of the frontiers of conjecture. 1878-1880-two years, famous lawyer and mathematician Kemp and Taylor, two were submitted demonstrate four color conjecture, announced that the four color theorem. But later mathematicians huwood to their own precise calculation that Kemp's proved to be wrong. Soon, Taylor proved also to be negative. So, people began to realize that the title of this seemingly easy, in fact, is a comparable with Fermat's conjecture. Since the beginning of the 20 th century, scientists conjecture on four color proof is basically the idea of according Kemp. The advent of electronic computer as a result of the rapid increase, calculus speed combined with the man-machine conversation, dramatically accelerating proof on four color process. conjecture In 1976, the United States mathematician Apel and Haken in United States, Illinois University of two different electronic computer, using the 1200 hours, made 100 million judgment, finally completed the four color theorem. However many mathematicians do not meet the achievements made in the computer, they think that there should be a simple and direct the written confirmation method. A few examples of the above mentioned are some and geometry-related issues, but these problems also with traditional geometry, but some new geometric concepts. These are the harbinger of "topology". What is topology? topology of the English name is a literal translation is to Topology, bibliography, and study of terrain, terrain similar to that of related subjects. Our early has translated into "the situation of geometry," and "continuous geometry," and "one on one continuous transformation groups of geometry", however, that several translation is not very good understanding of the unity of the 1956 mathematical terminology of identifying it as the topology, which is based on the transliteration. Topology is the geometry of a branch, but this geometry and and usual plane geometry, solid geometry. Usually of plane geometry or solid geometry of the object is a point, line and plane relationship between location and the nature of their metrics. Topology for the length of the object of study, size, area, volume measure the relationship between the nature and quantity are irrelevant. For example, in the normal plane geometry, the plane of a graphic to move to another graphic, if you complete heavy match, then the two graphics called whole, etc. However, in a study of topology is shown in the graph, in the campaign regardless of its size or shape is changed. In the topology, there is no rigid elements, each graphic, shape can change. For example, the preceding speaks of Euler in settlement of Fort VII bridge Yihong problems, he painted graphics won't consider its size, shape, only consider the number of dots and lines. These are the starting point topology thinking. Topological properties are those? first we introduced topology equivalent, which is easier to understand the nature of a topology. In topology doesn't discussing two graphic concept of the whole, however, discuss the concept of topological equivalent. For example, although the circle and square, triangular shape, size, topological transformation, they are equivalent graphic. Left of three things is topology equivalent, in other words, that is, from the point of view of topology, which is exactly the same. In a choice of some of the points on a sphere with a disjoint line connecting them, so that the lines are spherical was divided into many pieces. In topological transformation, points, lines, blocks of numbers are the same and the original number, this is equivalent to the topology. Generally, for any shape of closed surface, as long as the surface tear or cut his transformation is topology topology changes, there are equivalent. It should be noted that the torus does not have this property. Such as left, cut the torus, it should not be split into many pieces, only to become a bent round barrel, in this case, we say that the spherical topology cannot become a torus. So the spherical and toroidal the topology is different surfaces. Point on line and the line of binding relationship, the relationship between the order, in the topology of transformation, this is a topological properties. In topology of curves and surfaces of closed nature is topological properties. We usually speak of plane, surface usually has two sides, like a piece of paper has two sides. But Germany mathematician Moby Deus (1790-1868) in 1858 discovered Moby Deus surfaces. This surface will not be able to use different colors to fill two side. Topological transformation of invariance, not variables, there are many, here is not in the introduction. Topology creation, because other mathematics development needs, it has been rapid development. In particular, Riemann was founded after the Riemannian Geometry, he put the topology concepts as the basis for the analysis function is on, the more progress in the promotion of the topology. Since the beginning of the twentieth century, set theory was introduced topology for topology has opened up a new look. Topology of becomes arbitrary set of points corresponding to the concept. Topology in some need accurate description of the issue can be applied to the collection. Because a large number of natural phenomena have continuity, so contact the topology has a wide range of various practical things. Through a study of topology, you can illustrate the spatial structure of the collection, in order to master the functional relationship between the space. 1930s, mathematicians on the topology of more in-depth, made many new concepts. For example, consistent structure concept, an abstract from the concept and approximate space concept, and so on. There is a branch of mathematics called differential geometry, differential extraction line of tools to study at one point, surfaces in the vicinity of bent, and the topology is the study of global contact surface, so that the two subjects should exist some essence of the contact. In 1945, the American Chinese mathematician-Shen Chern established in algebraic topology and differential geometry, and advancing the overall development of the geometry. Topological till today, in theory, have very clearly divided into two branches. A branch is a strong bias towards analytical approach to the study, called the point set topology, or analysis of topology. Another branch is biased toward using algebraic methods to study, called algebraic topology. Now, these two branches and a consistent trend. Topology in functional analysis, lie theory, differential geometry, differential for many other branches of mathematics has a wide range of applications. (Transfers from big science network)