About the Book: This book provides exposition of the subject both in its general and algebraic aspects. It deals with the notions of topological spaces, compactness, connectedness, completeness including metrizability and compactification, algebraic aspects of topological spaces through homotopy groups and homology groups. It begins with the basic notions of topological spaces but soon going beyond them reaches the domain of algebra through the notions of homotopy, homology and cohomology. How these approaches work in harmony is the subject matter of this book. The book finally arrives at the realm of differentiable manifold and fibre bundles. It should be usefull for mathematicians as well as Physicists desiring to acquire overall knowledge of algebraic topology and its tools. Contents: PART I- Sets, Relations and Functions Topology of R and R2 Metric Space Topological Spaces Separation Axioms Compactness Connectedness PART II - Algebraic Preliminaries Homotopy Theory Compact Open Topology Higher Homotopy Groups Surfaces, Manifolds and CW-complexes Simplicial Homology Theory Singular Homology Theory Manifold Analysis Fibre Bundles.