A Simplified Book on Metric Spaces

This book is intended as a resource text for undergraduate students and a reference book for the subjects like topology at postgraduate level as the topological space are based on the properties of open sets in metric spaces. The subject of metric space is very interesting and it is the foundation course for the advance study in topology, algebraic topology and analysis. For the study of metric space, the reader should have a better understanding of a set theory. So, I have included the basics of sets, functions, relations and logic in the first chapter. I usually prefer to introduce continuity as early as possible, immediately after open sets and convergent sequences are introduced. I am of the opinion that open sets, convergence of sequences, continuity, compactness, connectedness and completeness should be the focus of attention and core of the course. I have chosen large number of examples which allow us to draw pictures and develop our intuition and draw conclusions, generate ideas for proofs. The aim of this course is to treat this as a preparatory ground for a general topology course and arm the reader with a repertory of examples.