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Although the topology has originated in different manners, it is a good idea to learn metric space first and then topology. Therefore the present book is completely dedicated to metric spaces and their topology. This subject is a necessity for many mathematical specialties and other fields. One is likely to approach the topic from their own perspective. The present book is an outcome of my lecture notes, which were prepared to teach topology at the Central University of Rajasthan, India. This book tries to answer the questions asked during the classroom teaching, either by me or by the students. The book starts with a basic fundamental question: “What is a set?” Why an axiomatic system is required. In the first chapter, the Zermelo–Freenkel axiomatic system is discussed. The complete description of the various number systems has been discussed in the first chapter. The constructions of the real number system with the help of Dedekind cuts and Cauchy sequences are discussed in the Chapters 1 and 5, respectively. Chapter 2 starts with the definition of a metric space. In this chapter, various examples and properties of the metric spaces and normed spaces are discussed. Also, the distances between the sets in metric spaces and the Gromov–Hausdorff distance are discussed. Chapter 3 is devoted to maps between metric spaces. The first section discusses continuous maps and their properties. The second section discusses homeomorphisms and their properties. The third section discusses equivalent metrics. The notions of Lipschitz equivalent and topological equivalent metrics are discussed. In the fourth section the isometry between metric spaces is discussed. The fifth section discusses the finite metric spaces and their embedding in Euclidean spaces. In Chapter 4, metrics on the product and quotient sets are discussed. Again, the metric-preserving maps are discussed in this setting. Chapter 5 is devoted to sequences in metric spaces. The first section deals with convergent sequences and their properties. The second section discusses complete metric spaces and their properties. In the fifth section the completion of a metric space is discussed. In the fourth section the real number system is obtained via the completion of the rationals, which is an incomplete metric space. In the fifth section, the Baire category theorem and its applications are discussed. Chapter 6 is devoted to compact metric spaces. The first section discusses their properties. In the second section, some equivalence of compactness is discussed. In the third section the Hilbert cube is discussed, and it is proved that a compact metric space can be embedded in the Hilbert cube as a closed set. The fourth section discusses the Cantor set, and it is shown that a compact metric space is a continuous image of the Cantor set. The last chapter is about connected metric spaces. Various properties of connected spaces are discussed in the first and second sections. The last section discusses the pathconnected metric spaces. A book cannot be completed without the help of others. First of all, I am thankful to my teachers, Prof. Ramji Lal (retired) and Prof. R. P. Shukla, University of Allahabad, Prayagraj, for their constant encouragement. I am also grateful to Prof. D. P. Choudhary (retired), University of Allahabad, Prayagraj, for pointing out various errors in this book. However, if there are any errors left, I am the only one responsible for them. I will be thankful to the readers if they can let me know any kind of errors in the book. I am very thankful to the editorial team of De Gruyter, especially Dr. Ranis N. Ibragimov. He was very helpful at various stages. I am thankful to my parents, Mr. V. K. Khattri and Mrs. Gaytri Khattri, for their constant moral support. Last but not least, my wife Vinny has my sincere gratitude for her patience, unwavering support, and encouragement all across my life.