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2022 | Buch

Basic Topology 1

Metric Spaces and General Topology

verfasst von: Prof. Avishek Adhikari, Prof. Dr. Mahima Ranjan Adhikari

Verlag: Springer Nature Singapore


Über dieses Buch

This first of the three-volume book is targeted as a basic course in topology for undergraduate and graduate students of mathematics. It studies metric spaces and general topology. It starts with the concept of the metric which is an abstraction of distance in the Euclidean space. The special structure of a metric space induces a topology that leads to many applications of topology in modern analysis and modern algebra, as shown in this volume. This volume also studies topological properties such as compactness and connectedness. Considering the importance of compactness in mathematics, this study covers the Stone–Cech compactification and Alexandroff one-point compactification. This volume also includes the Urysohn lemma, Urysohn metrization theorem, Tietz extension theorem, and Gelfand–Kolmogoroff theorem.

The content of this volume is spread into eight chapters of which the last chapter conveys the history of metric spaces and the history of the emergence of the concepts leading to the development of topology as a subject with their motivations with an emphasis on general topology. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power, and active learning of the subject, all the while covering a wide range of theories and applications in a balanced unified way.


Chapter 1. Prerequisites: Sets, Algebraic Systems and Classical Analysis
This chapter assembles together some basic concepts and results of set theset theory, modern algebra, classical analysis and also of category theory by using a natural language for smooth reading of the book.
Avishek Adhikari, Mahima Ranjan Adhikari
Chapter 2. Metric Spaces and Normed Linear Spaces

This chapter starts a journey in metric spaces describing the concept of metrics, which is an abstraction of distance in the Euclidean space and conveys an axiomatic framework for this abstraction with a systemic study of elementary basic properties of metric spaces. It also discusses normed linear spaces which form a versatile class of metric spaces. This discussion includes a brief study of Banach and Hilbert spaces.

Avishek Adhikari, Mahima Ranjan Adhikari
Chapter 3. Topological Spaces and Continuous Maps

The subject Topology sets out its official journey in this chapter through the address of the concepts of topological spaces and their continuous maps, which are the basic objects and the basic functions in topology.

Avishek Adhikari, Mahima Ranjan Adhikari
Chapter 4. Separation Axioms
This chapter studies topological spaces by imposing certain conditions, called separation axioms on these spaces in terms of their points and open sets, specially, where there is possibly no concept of distance. The additional conditions are needed, because the defining axioms for a topological space are extremely general and they are too weak to study them in depth.
Avishek Adhikari, Mahima Ranjan Adhikari
Chapter 5. Compactness and Connectedness

This chapter is devoted to address the concepts of compactness and connectedness in topological settings, which first arose through the study of subsets of the Euclidean line \( \mathbf {R}\) in calculus and mathematical analysis.

Avishek Adhikari, Mahima Ranjan Adhikari
Chapter 6. Real-Valued Continuous Functions
This chapter continues the study of continuous functions from a topological space to the real line space \( \mathbf{R}, \) called the real-valued continuous functions, or, simply, real functions; such functions play a central role in topology and analysis. This chapter also studies uniform convergence of real-valued functions and characterizes normal spaces through separation by real-valued continuous functions.
Avishek Adhikari, Mahima Ranjan Adhikari
Chapter 7. Countability, Separability and Embedding
This chapter continues the study of special classes of topological spaces such as spaces satisfying either of the two axioms of countability formulated by Hausdorff in 1914 or satisfying the axiom of separability introduced by Frechét in 1906, both initiated in Chap. 3, which do not arise from the study of calculus and analysis in a natural way. They arise through a deep study of topology. The axiom of first countability arose through the study of convergent sequences.
Avishek Adhikari, Mahima Ranjan Adhikari
Chapter 8. Brief History of Topology I: Motivation of the Subject with Historical Development
The subject Topology has become one of the most exciting and influential fields of study in modern mathematics, because of its beauty and scope. Topology starts where sets have some cohesive properties leading to define continuity of functions.
Avishek Adhikari, Mahima Ranjan Adhikari
Basic Topology 1
verfasst von
Prof. Avishek Adhikari
Prof. Dr. Mahima Ranjan Adhikari
Springer Nature Singapore
Electronic ISBN
Print ISBN