Introduction 1. What is point-set topology about? 2. Origin and beginnings CHAPTER I Fundamental Concepts l. The concept of a topological space 2. Metric spaces 3. Subspaces, disjoint unions and products 4. Bases and subbases 5. Continuous maps 6. Connectedness 7. The Hausdorff separation axiom 8. Compactness CHAPTER II Topological Vector Spaces 1. The notion of a topological vector space 2. Finite-dimensional vector spaces 3. Hilbert spaces 4. Banach spaces 5. Frechet spaces 6. Locally convex topological vector spaces 7. A couple of examples CHAPTER III The otient Topology 1. The notion of a quotient space 2. otients and maps 3. Properties of quotient spaces 4. Examples: Homogeneous spaces 5. Examples: Orbit spaces 6. Examples: Collapsing a subspace to a point 7. Examples: Gluing topological spaces together CHAPTER IV Comlio of Metric Spaces 1. The comlio of a metric space 2. Comlio of a map 3. Comlio of normed spaces CHAPTER V Homotopy I. Homotopic maps 2. Homotopy equivalence 3. Examples 4. Categories 5. Functors 6. What is algebraic topology? 7. Homotopy--what for? CHAPTER VI The Two Countability Axioms 1. First and second countability axioms 2. Infinite products 3. The role of the countability axioms CHAPTER VII CW-Complexes 1. Simplicial complexes 2. Cell decoitions 3. The notion of a CW-complex 4. Subcomplexes 5. Cell attaching