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 Quotient Topology
1. The notion of a quotient space
2. Quotients 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 Completion of Metric Spaces
1. The completion of a metric space
2. Completion of a map
3. Completion 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 decompositions
3. The notion of a CW-complex
4. Subcomplexes
5. Cell attaching
