Book:Klaus Jänich/Topology

Klaus Jänich: Topology

Published $\text {1984}$, Springer

ISBN 978-3540908920

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—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