Book:Klaus Jänich/Topology
Jump to navigation
Jump to search
Klaus Jänich: Topology
Published $\text {1984}$, Springer
- ISBN 978-3540908920
Subject Matter
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—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