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In the field of mathematics, topology is the study of the structure of space, which eschews the details of specific shapes and focuses on the study of essential features and interrelationships within space. In contrast, narrow geometry focuses more on the study of shape. In this article, we will introduce you to the basic concepts and preparatory knowledge of topology to help you take your first steps in learning topology.
- Topology: forget about specific information about shapes and study more essential spatial structures
- Geometry in a narrow sense: the study of shapes
Preliminaries
Some preparatory knowledge is necessary before studying topology.
Sets and mappings
- A set is a whole composed of some specific elements. We can perform basic set operations such as intersection, union, complement and difference.
- Mapping describes the correspondence between two sets. One-shot, full-shot and bijection are common concepts of mapping. In addition, original image and image are important concepts that describe the behavior of mappings.
Calculus
- Limits are a central concept in calculus that describes the behavior of a function near a certain point.
- Continuous functions are a class of functions that have no jumps or discontinuities throughout their domain of definition.
A little bit of abstract algebra
- A group is an algebraic structure that describes a set with specific rules of operation. The study of groups can help us understand the symmetry of some topological structures.
- Homomorphisms are maps that preserve the structure of a group and they play an important role in group theory.
- A quotient group is a new group constructed by dividing a group by equivalence relations, which categorizes the elements of the original group according to equivalence relations.
Bibliography
The following books are excellent references in learning topology:
- Basic Topology (by Armstrong) This book provides a systematic introduction to the basic concepts and methods of topology and is ideal for beginners. It has a Chinese translation, Basic Topology, which can also be used as a reference.
- Topology (Author: Munkres) Munkres' Topology is a classic textbook that provides an in-depth introduction to the basics and applications of topology.
- Galois groups and fundamental groups (by Szamuely)
Combining topology with algebra, this book delves into concepts such as Galios groups and fundamental groups, and is very helpful in understanding the algebraic characterization of topological spaces.
- Algebraic Topology (by Tom Dieck) Algebraic Topology is an advanced topology textbook that covers a wide range of algebraic topology, including important concepts such as cohomology and homology.
- Elements of Algebraic Topology (by Munkres) This book provides a clear and detailed introduction to the basic concepts of algebraic topology for beginners.
Reference Tutorials
If you are interested in basic topology, the following tutorials can help you learn more:
- Basic Topology: 60 Hours of Learning Tutorials This tutorial provides you with a comprehensive and systematic path to basic topology. With ample exercises and case studies, you will gradually master the core concepts and techniques of topology.
Through in-depth study of the above content, as well as continuous practice and exploration, you will gradually master the core ideas and methods of topology. Topology, as an abstract and subtle discipline, will open a door to the world of mathematics for you.
- 作者:Liang
- 链接:https://www.liangwatcher.top/article/tuopujc1
- 声明:本文采用 CC BY-NC-SA 4.0 许可协议，转载请注明出处。
