Book Introduction

This book systematically introduces the basic concepts and properties of point set topology, covering the properties of mapping: metric space and completeness; Equivalence characterization of open set, neighborhood, closure, interior, boundary, base and sub-base in topological space. Equivalent conditions of continuous mapping, open-closed mapping and homeomorphism mapping; The relationship between net and filter and its convergence; Subspaces, product spaces and quotient spaces of topological spaces; Connectivity, local connectivity, road connectivity and their topological properties; Countability, separability, Ti (I = 0, 1, 2, 3, 4, 5) separability, regularity and normal separability, Urysohn separability, complete regularity and complete normal separability; Compactness, local compactness and paracompactness and their applications: conditions of compact metric space, measurable topological space, generalized open (closed) set, generalized continuous mapping, etc.

This book is rich in content, novel in theory, clear in thinking and easy to understand. It is suitable for reading reference for senior undergraduates and graduate students, and can also be used as teaching reference for teachers and scientific and technological personnel in related fields.

This book has eight chapters. The first chapter is the foundation, which introduces the concepts, symbols and terms of sets and set families, relations, mappings and their properties. The second chapter introduces the basic concepts and properties of metric space, including the neighborhood of points, the properties of open sets,

Sequence, convergence and completeness of metric spaces. The purpose is to prepare for introducing topological space and related concepts, making topological space measurable, and making it easier for beginners to understand the concept of topological space. Chapters 3 to 6 are the core parts of point set topology, including the equivalent descriptions of open set, neighborhood, closure, interior, boundary, basis and subbase, and the equivalent conditions of continuous mapping, open-close mapping and homeomorphism mapping in topological space. The relationship between net and filter and its convergence; Topological properties of subspace, product space and quotient space of topological space; Connectivity and its inheritance, multiplication and topological invariance, local connectivity, road connected space and its application, topological invariance; The first countable and second countable separable spaces, Hausdorff separability, regularity, normal separability, Urysohn separability, complete regularity, and the heritability, multiplicability, topological invariance and complete normal separability of these separability; Compact space, countable compactness, convergent compactness, local compactness, paracompact compactness and compactness, and their relations in separability, etc. Chapter 7: Conditions of compact metric space and measurable topological space. Chapter 8 is the closed graphs of generalized open-closed sets and generalized continuous, continuous and almost continuous mappings. Most of them are the latest achievements in recent years.

There are beginners and advanced ones.