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The branch of mathematics that studies continuity. Topology studies the invariance of topological space under homeomorphism (topology) transformation. Homeomorphic transformation refers to continuous transformation, and its inverse mapping is also continuous. Intuitively speaking, it is to study the unchangeable properties of graphics under this deformation: graphics can be bent, widened or contracted at will, as long as the origin does not stick to a point, no new points will be generated during deformation. The word topology is a combination of two Greek words, Topos, which means position, and Logos, which means idea. It was first used in J.B. Christine's Preliminary Topology (1847).
Topology came into being in19th century. It was divided into two branches. One is point set topology, which is produced on the basis of set theory for the needs of basic work. The second is combinatorial topology, which is produced for the need of geometry research. The former evolved into a general topology. The latter evolved into algebraic topology and geometric topology. Differential topology came into being in the 20th century.
Surface model with genus 1
The original topological theorem is "Solution of the Seven Bridges in Konigsberg" published by L. Euler in 1736, which gives the necessary and sufficient conditions for the network to be connected. In 1750, he gave euler theorem's assertion that the number of vertices a0, the number of edges a 1 and the number of faces a2 of a polyhedron which is homeomorphic to a two-dimensional spherical surface S2 (that is, it can be continuously transformed into S2) satisfy A0-A 1+A2 = 2, which is the Euler characteristic of S2. There are only five kinds of convex regular polyhedra that can be proved: regular tetrahedron, regular octahedron, regular dodecahedron, regular hexahedron and regular icosahedron. It plays an important role in the classification of closed surfaces and the proof of four-color problems. After Euler, around 1833, C.F. Gauss studied all kinds of knots in space (that is, knotted non-self-closing curves) to see if knots could be opened. Whether two knots can deform each other is an equivalent classification of knots. Gauss gives the number of cycles of a closed curve, which is one of the basic tools for the study of kink theory. Are the research results of the embryonic stage of topology.
1895 B. Riemann put forward the concept of Riemann surface. When he was studying complex variable functions, he began to systematically study the topological properties of surfaces. The classification problem of directed closed surfaces is solved, and the precise definition of n-dimensional manifold is given.
J.-H. Poincare is recognized as the founder of combinatorial topology. 1895 published several papers entitled "On Location Geometry", which was the beginning of the study of topological space algebra.
Another branch of topology, point set topology, originated from the basic work of mathematical analysis and the generation of functional analysis. At the end of 19, G. Cantor established set theory, further defined open set, closed set and derived set of Euclidean space, and obtained important results of topological structure of Euclidean space. The rise of functional analysis later promoted the study of point set as space.
Differential topology originated from Poincare conjecture, and began to form an independent branch of mathematics after H. Whitney published the embedding theorem in 1936.
Topology permeates all branches of mathematics, just as the study of homology groups and homotopy groups promotes the development of homology algebra; The study of fiber bundle, differential manifold and differential topology promotes the development of differential geometry; Differential dynamic system is an interdisciplinary subject of differential topology and differential equations. Moreover, topology has been widely used in physics, chemistry, biology, economics and other disciplines. One example is the application of knots in physics and genetic engineering.