Chapter-1: Introduction; 1. Meaning of Topology; 2. Nature of Topology; 3. Topological Equivalence or Homeomorphism; 4. The Mobious band and one-sided surface; 5. Historical Aspects; Chapter-2: Preliminary Concepts of Sets and Functions: 1. Introduction to Set Theory; 2. Introduction to the Theory of Functions ( or Mappings ); 3. Relations; 4. Order and Lattices etc.; 5. Real points, Cardinality and Countability; 6. Bounded, Derived, Closed , Open and Compact Sets, Closure of a Set; 7. Cover and Compact Set in terms of Cover; 8. Sequences; 9. Algebric Structures; Chapter-3: Metric Spaces: 1. Introductions; 2. Open Sets in a Metric Space; 3. Interior, Exterior and Frontier Points; 4. Closed Sets in a metric Space; 5. Convergence and Completeness of Metric Spaces; 6. Some other Definitions and Spaces related to Metric Space; 7. Contability and Separation Properties; 8. Continuity and Homeomorphism in Metric Space, Limit of a Function; 9. Completion of a Metric Space; Chapter-4: Topological Spaces: 1. Introduction; 2. Definition of Topologies; 3. Sets in Topological Space; 4. Topology Induced by the Matric or the Matric Topology or Metrization; 5. Mappings, Continuity and Homeomorphism of Topological Space; Chapter-5: Seperation Properties of Topological Spaces: 13. Introduction; 14. Spaces or Kolmogo;