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Contents: Introduction. - Fundamental Concepts. -
Topological Vector Spaces.- The Quotient Topology. -
Completion of Metric Spaces. - Homotopy. - The Two
Countability Axioms. - CW-Complexes. - Construction of
Continuous Functions on Topological Spaces. - Covering
Spaces. - The Theorem of Tychonoff. - Set Theory (by T.
Br cker). - References. - Table of Symbols. -Index.

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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. Fréchet 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 Bröcker).- References.- Table of Symbols.

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