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
6. Why CW-complexes are more flexible 7. Yes, but... ? CHAPTER VIII Construction of Continuous Functions on Topological Spaces 1. The Urysohn lemma 2. The proof of the Urysohn lemma 3. The Tietze extension lemma 4. Partitions of unity and vector bundle sections 5. Paracompactness CHAPTER IX Covering Spaces 1. Topological spaces over X 2. The concept of a covering space 3. Path lifting 4. Introduction to the classification of covering spaces 5. Fundamental group and lifting behavior 6. The classification of covering spaces 7. Covering transformations and universal cover 8. The role of covering spaces in mathematics CHAPTER X The Theorem of Tyehonoff 1. An unlikely theorem? 2. What is it good for? 3. The proof LAST CHAPTER Set Theory (by Thcodor Brfcker) References Table of Symbols Index
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