1. Topology is an important basic subject of mathematics, which together with algebra constitutes two pillars of mathematics. If algebra studies the general theory of discrete operations, then topology studies the general theory of continuous mapping. Compared with other branches of mathematics, topology is a young discipline, which was crystallized into a branch of geometry in the early 20th century from the development of19th century. Topology studies the property that geometric figures remain unchanged after arbitrary deformation. These deformations can be compression, stretching or arbitrary bending, etc. However, in the process of deformation, it is not allowed to create new points or combine two points. In other words, the adjacent points of the graph are still adjacent after deformation, which is called continuity; In addition, there is a one-to-one correspondence between graphics and deformation points. Therefore, this deformation is required to be continuous and the inverse transformation is also continuous. This transformation is called topological equivalence or homeomorphism. Topology has an image nickname-rubber geometry, because if the graphics are made of rubber, many graphics can become homeomorphic graphics.

Topology has many different origins, which makes it divided into several branches, mainly point set topology and algebraic topology. Point set topology, also known as general topology, was formed under the strong influence of Cantor's set theory. It originated from Frechet 1906' s paper on general metric space theory and Hausdorff 19 12' s book The Basis of Set Theory.

The introduction of Hilbert space, Banach space and the rise of functional analysis show the importance of introducing abstract point sets into appropriate structures and studying them as spaces. Topological space is such a set, which is endowed with a certain structure. With this structure, we can talk about the proximity between points or subsets, and then we can talk about the continuity of mapping. In classical analysis and functional analysis, the limit of sequence plays an important role, so those properties that play a role in analysis are topological properties. Operators in functional analysis are mappings from one space to another. Therefore, topology naturally becomes a tool to study functional analysis. The origin of algebraic topology is different from that of point set topology, and its history can be traced back to a longer time. Euler theorem on the polyhedron has seen the clue of algebraic topology. Euler was interested in this theorem because he wanted to use it to classify polyhedrons. But he didn't notice the invariance under continuous transformation. The classification of surfaces and Riemann's theory of complex variable function are both aimed at promoting topology. He introduced fundamental groups and homology groups. What prompted him to study topology was some classical geometric problems and integral theory. Topological methods and many concepts have penetrated into almost all fields of mathematics, and have been applied in physics, chemistry, biology and other disciplines, and these applications will be more extensive in the future. Huazhang Mathematical Topology (2nd Edition)/Translation Series

Author: (America) Munchris Translator: Xiong Jincheng Lv Jie Tan Feng

2. Euclidean space is the generalization of two-dimensional and three-dimensional space studied by Euclid in mathematics. This generalization transforms Euclid's concepts of distance and related length and angle into a coordinate system of any dimension. This is a "standard" example of finite dimension, real number and inner product space.

Euclidean space is a special metric space, which enables us to study its topological properties, such as compactness. Inner product space is a generalization of Euclidean space. Both inner product space and metric space are discussed in functional analysis.

Euclidean space plays a role in the definition of manifolds containing Euclidean geometry and non-Euclidean geometry. The mathematical motivation of defining distance function is to define the tee-off around a point in space. This basic concept proves the difference between Euclidean space and other manifolds. Differential geometry introduces differentiation, transfer skills and local Euclidean space, and discusses the properties of non-Euclidean manifolds.

Topology is an important branch of modern mathematics, which mainly studies the laws of singular deformation. Generally speaking, topology is mathematics on rubber: after drawing a regular figure (such as a rectangular grid) on elastic rubber, twist it by hand at will, and all kinds of strange changes will happen to the figure drawn on it, and you will find wonderful figures that you have never seen before; Or you can pinch a balloon with your hand and let it expand, and you will see incredible changes in the patterns printed on it. Topology is used to study the law of the beauty of this graphic change.