Preface 1 Theory of Sets 1 Introduction 2 Sets and subsets 3 "Set operations: union, intersection, and complement" 4 Indexed families of sets 5 Products of sets 6 Functions 7 Relations 8 Composition of functions and diagrams 9 "Inverse functions, extensions, and restrictions" 10 Arbitrary products 2 Metric Spaces 1 Introduction 2 Metric spaces 3 Continuity 4 Open balls and neighborhoods 5 Limits 6 Open sets and closed sets 7 Subspaces and equivalence of metric spaces 8 An infinite dimensional Euclidean space 3 Topological Spaces 1 Introduction 2 Topological spaces 3 Neighborhoods and neighborhood spaces 4 "Closure, interior, boundary" 5 "Functions, continuity, homeomorphism" 6 Subspaces 7 Products 8 Identification topologies 9 Categories and functors 4 Connectedness 1 Introduction 2 Connectedness 3 Connectedness on the real line 4 Some applications of connectedness 5 Components and local connectedness 6 Path-connected topological spaces 7 Homotopic paths and the fundamental group 8 Simple connectedness 5 Compactness 1 Introduction 2 Compact topological spaces 3 Compact subsets of the real line 4 Products of compact spaces 5 Compact metric spaces 6 Compactness and the Bolzano-Weierstrass property 7 Surfaces by identification Bibliography Index
