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This book is an ideal resource for students embarking on their first course in topology, particularly those pursuing an M.Sc. in Mathematics. The author, drawing upon extensive experience teaching topology for many years, has meticulously crafted the content to address common questions and clarify typical doubts students encounter.
Key Features
• Student-Centric Approach: The explanations of proofs and examples are directly informed by the real-world questions and uncertainties observed in the classroom. This ensures the material directly tackles the conceptual hurdles students often face. • Accessible Examples: Unlike texts that might overwhelm with overly complex scenarios, this book focuses on simple, illustrative examples. This approach prioritizes understanding fundamental concepts over grappling with intricate details, making the subject more approachable for beginners. • Appropriate Level of Detail: The content is carefully curated to be sufficient for a typical M.Sc. Mathematics student without delving into overly advanced or specialized topics. This focused scope helps students build a strong foundational understanding of topology without being intimidated by extraneous material.
In essence, this book acts as a patient and insightful guide, anticipating student difficulties and providing clear, concise, and relevant explanations that foster a solid grasp of topological principles.
Dr. Prabhakar is serving as an Associate Professor at the Department of Mathematics, IIT Ropar. Renowned for his engaging and effective teaching style, Dr. Prabhakar is widely regarded as one of the most popular faculty members at IIT Ropar. His expertise and passion for mathematics have inspired countless students throughout his career. After completing his doctoral studies at IIT Delhi, Dr. Prabhakar further honed his research skills through postdoctoral work at the Harish Chandra Research Institute, Allahabad. His academic journey also included a fruitful stint as a visiting research fellow at Osaka City University, Japan. Prior to joining IIT Ropar, Dr. Prabhakar served as a faculty member at IIT Guwahati for three years. With over 18 years of dedicated teaching and research, he has made significant contributions to the field of Knot Theory, a specialized area within low-dimensional topology. His research has been recognized through numerous publications in esteemed international journals.
Preface
1 Introduction
1.1 Topological Spaces
1.2 Subspace Topology
1.3 Basis for a Topology
1.4 Subbasis for a Topology
1.5 Exercises
2 Closed sets and Some Properties
2.1 Closed Sets, limit and interior points
2.2 Hausdorff Spaces
2.3 First Countable Space
2.4 Continuous Functions and Homeomorphisms
2.5 Exercises
3 New Classes of Topological Spaces
3.1 Finite Product Topology
3.2 Infinite Product Topology
3.3 Metrizable Spaces
3.4 Quotient Spaces
3.5 Exercises
4 Connectedness
4.1 Connected Spaces
4.2 Path-Connected Spaces
4.3 Components and Path Components
4.4 Locally Connected and Locally Path-Connected Spaces
4.5 Exercises
5 Compactness
5.1 Compact Spaces
5.2 Products of Compact Spaces
5.3 Compactness in metric spaces
5.4 Limit Point Compactness and Sequentially Compactness
5.5 Local Compactness
5.6 Exercises
6 Countability & Separation Axioms
6.1 The Countability Axioms
6.2 Separation Axioms
6.3 The Urysohn Lemma
6.4 The Tietze Extension Theorem
6.5 The Urysohn Metrization Theorem
6.6 Exercises
A Sets and Functions
A.1 Sets
A.2 Relations and Functions
A.3 Finite and Infinite Sets
A.4 Cartesian Products of Sets
References
Index
