Elementary Topology

Textbook in Problems

by O.Viro, O.Ivanov, V.Kharlamov, N.Netsvetaev

Table of Contents

• Introduction

• Contents
• General Topology
• Structures and Spaces
• Digression on Sets
• Topology in a Set
• Bases
• Metric Spaces
• Subspaces
• Position of a Point with Respect to a Set
• Ordered Sets
• Continuity
• Set-Theoretic Digression. Maps
• Continuous Maps
• Homeomorphisms
• Topological Properties
• Connectedness
• Applications of Connectedness
• Path-Connectedness
• Separation Axioms
• Countability Axioms
• Compactness
• Sequential Compactness
• Local Compactness and Paracompactness
• Topological Constructions
• Multiplication
• Quotient Spaces
• Zoo of Quotient Spaces
• Projective Spaces
• Finite Topological Spaces
• Spaces of Continuous Maps
• Topological Algebra
• Digression. Generalities on Groups
• Topological Groups
• Consructions
• Actions of Topological Groups
• Algebraic Topology
• Fundamental Group
• Homotopy
• Homotopy Properties of Path Multiplication
• Fundamental Group
• The Role of Base Point
• Covering Spaces and Calculation of Fundamental Groups
• Covering Spaces
• Theorems on Path Lifting
• Calculations of Fundamental Groups Using Universal Coverings
• Fundamental Group and Mappings
• Induced Homomorphisms and Their First Applications
• Retractions and Fixed Points
• Homotopy Equivalences
• Covering Spaces via Fundamental Groups
• Cellular Techniques
• Cellular Spaces
• Cellular Constructions
• One-Dimensional Cellular Spaces
• Fundamental Group of Cellular Space
• Hints, Comments, Advises, Solutions, and Answers

2007-07-13