Aniruddha Deshmukh & Dhananjay Gopal 
Introduction to Metric Spaces [PDF ebook] 

This book serves as a textbook for an introductory course in metric spaces for undergraduate or graduate students. The goal is to present the basics of metric spaces in a natural and intuitive way and encourage students to think geometrically while actively participating in the learning of this subject. In this book, the authors illustrated the strategy of the proofs of various theorems that motivate readers to complete them on their own. Bits of pertinent history are infused in the text, including brief biographies of some of the central players in the development of metric spaces. The textbook is divided into seven chapters that contain the main materials on metric spaces; namely, introductory concepts, completeness, compactness, connectedness, continuous functions and metric fixed point theorems with applications.Some of the noteworthy features of this book include* Diagrammatic illustrations that encourage readers to think geometrically* Focus on systematic strategy to generate ideas for the proofs of theorems* A wealth of remarks, observations along with a variety of exercises* Historical notes and brief biographies appearing throughout the text