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1. What is point-set topology about?.- 2. Origin and beginnings.- I Fundamental Concepts.- 1. 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.- 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.- 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.- IV Completion of Metric Spaces.- 1. The completion of a metric space.- 2. Completion of a map.- 3. Completion of normed spaces.- V Homotopy.- 1. Homotopic maps.- 2. Homotopy equivalence.- 3. Examples.- 4. Categories.- 5. Functors.- 6. What is algebraic topology?.- 7. Homotopy-what for?.- VI The Two Countability Axioms.- 1. First and second countability axioms.- 2. Infinite products.- 3. The role of the countability axioms.- 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... ?.- 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.- 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.- X The Theorem of Tychonoff.- 1. An unlikely theorem?.- 2. What is it good for?.- 3. The proof.- Last Chapter Set Theory (by Theodor Broecker).- References.- Table of Symbols.

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