Introduction to Topology

Description

The aim of the book is to give a broad introduction of topology to undergraduate students. It covers the most important and useful parts of the point-set as well as the combinatorial topology. The development of the material is from simple to complex, concrete to abstract, and appeals to the intuition of readers. Attention is also paid to how topology is actually used in the other fields of mathematics. Over 150 illustrations, 160 examples and 600 exercises will help readers to practice and fully understand the subject. Contents: Set and Map Metric Space Graph Topology Topological Concepts Complex Topological Properties Surface Topics in Point Set Topology Index Covers point set topology and combinatorial topology Contains abundant examples and exercises to facilitate the study Entry level textbook, allowing easy access to the topic Content 1. Set and Map 1.1 Set 1.2 Map 1.3 Counting 1.4 Equivalence Relation and Quotient 2. Metric Space 2.1 Metric 2.2 Ball 2.3 Open Subset 2.4 Continuity 2.5 Limit Point 2.6 Closed Subset 3. Graph 3.1 Seven Bridges in K¨onigsberg 3.2 Proof of One-Trip Criterion 3.3 Euler’s Formula 3.4 Application of Euler’s Formula 4. Topology 4.1 Topological Basis and Subbasis 4.2 Open Subset 4.3 Topological Space 4.4 Comparing Topologies 4.5 Limit Point and Closed Subset 4.6 Closure 5. Topological Concepts 5.1 Continuity 5.2 Homeomorphism 5.3 Subspace 5.4 Product 5.5 Quotient 6. Complex 6.1 Simplicial Complex 6.2 CW-Complex 6.3 Projective Space 6.4 Euler Number 7. Topological Properties 7.1 Hausdorff Space 7.2 Connected Space 7.3 Path Connected Space 7.4 Connected Component 7.5 Compact Space 7.6 Limit Point Compact Space 8. Surface 8.1 Manifold 8.2 Surface 8.3 Simplicial Surface 8.4 Planar Diagram 8.5 Cut and Paste 8.6 Classification of Surface 8.7 Recognition of Surface 9. Topics in Point Set Topology 9.1 Normal Space 9.2 Paracompact Space 9.3 Complete Metric Space 9.4 Baire Category Theorem 9.5 Infinite Product 9.6 Space-Filling Curve 9.7 Space of Maps Index