# Book:Klaus Jänich/Topology

## Klaus Jänich: Topology

Published $1984$, Springer

ISBN 978-3540908920.

### 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