Book:Klaus Jänich/Topology

Subject Matter

 * Topology

Contents

 * Introduction
 * $\S 1$. What is point-set topology about?
 * $\S 2$. Origin and beginnings


 * CHAPTER I: Fundamental Concepts
 * $\S 1$. The concept of a topological space
 * $\S 2$. Metric spaces
 * $\S 3$. Subspaces, disjoint unions and products
 * $\S 4$. Bases and subbases
 * $\S 5$. Continuous maps
 * $\S 6$. Connectedness
 * $\S 7$. The Hausdorff separation axiom
 * $\S 8$. Compactness


 * CHAPTER II: Topological Vector Spaces
 * $\S 1$. The notion of a topological vector space
 * $\S 2$. Finite-dimensional vector spaces
 * $\S 3$. Hilbert spaces
 * $\S 4$. Banach spaces
 * $\S 5$. Fréchet spaces
 * $\S 6$. Locally convex topological vector spaces
 * $\S 7$. A couple of examples


 * CHAPTER III: The Quotient Topology
 * $\S 1$. The notion of a quotient space
 * $\S 2$. Quotients and maps
 * $\S 3$. Properties of quotient spaces
 * $\S 4$. Examples: Homogeneous spaces
 * $\S 5$. Examples: Orbit spaces
 * $\S 6$. Examples: Collapsing a subspace to a point
 * $\S 7$. Examples: Gluing topological spaces together


 * CHAPTER IV: Completion of Metric Spaces
 * $\S 1$. The completion of a metric space
 * $\S 2$. Completion of a map
 * $\S 3$. Completion of normed spaces


 * CHAPTER V: Homotopy
 * $\S 1$. Homotopic maps
 * $\S 2$. Homotopy equivalence
 * $\S 3$. Examples
 * $\S 4$. Categories
 * $\S 5$. Functors
 * $\S 6$. What is algebraic topology?
 * $\S 7$. Homotopy&mdash;what for?


 * CHAPTER VI: The Two Countability Axioms
 * $\S 1$. First and second countability axioms
 * $\S 2$. Infinite products
 * $\S 3$. The role of the countability axioms


 * CHAPTER VII: CW-Complexes
 * $\S 1$. Simplicial complexes
 * $\S 2$. Cell decompositions
 * $\S 3$. The notion of a CW-complex
 * $\S 4$. Subcomplexes
 * $\S 5$. Cell attaching
 * $\S 6$. Why CW-complexes are more flexible
 * $\S 7$. Yes, but. . . ?


 * CHAPTER VIII: Construction of Continuous Functions on Topological Spaces
 * $\S 1$. The Urysohn lemma
 * $\S 2$. The proof of the Urysohn lemma
 * $\S 3$. The Tietze extension lemma
 * $\S 4$. Partitions of unity and vector bundle sections
 * $\S 5$. Paracompactness


 * CHAPTER IX: Covering Spaces
 * $\S 1$. Topological spaces over X
 * $\S 2$. The concept of a covering space
 * $\S 3$. Path lifting
 * $\S 4$. Introduction to the classification of covering spaces
 * $\S 5$. Fundamental group and lifting behavior
 * $\S 6$. The classification of covering spaces
 * $\S 7$. Covering transfonnations and universal cover
 * $\S 8$. The role of covering spaces in mathematics


 * CHAPTER X: The Theorem of Tychonoff
 * $\S 1$. An unlikely theorem?
 * $\S 2$. What is it good for?
 * $\S 3$. The proof


 * LAST CHAPTER: Set Theory (by Theodor Bröcker)


 * References
 * Table of Symbols
 * Index