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| 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 |
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