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This is an introductory textbook on general and algebraic topology, aimed at anyone with a basic knowledge of calculus and linear algebra. It provides full proofs and includes many examples and exercises. The covered topics include: set theory and cardinal arithmetic; axiom of choice and Zorn's lemma; topological spaces and continuous functions; con- nectedness and compactness; Alexandrov compactification; quotient topol- ogies; countability and separation axioms; prebasis and Alexander's theorem; the Tychonoff theorem and paracompactness; complete metric spaces and function spaces; Baire spaces; homotopy of maps; the fundamental group; the van Kampen theorem; covering spaces; Brouwer and Borsuk's theorems; free groups and free product of groups and basic category theory. While it is very concrete at the beginning, abstract concepts are gradually introduced.
This second edition contains a new chapter with a topological introduction to sheaf cohomology and applications.It also corrects some inaccuracies and some additional exercises are proposed.
The textbook is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.
List of contents
1 Geometrical introduction to topology.- 2 Sets.- 3 Topological structures.- 4 Connectedness and compactness.- 5 Topological quotients.- 6 Sequences.- 7 Manifolds, infinite products and paracompactness.- 8 More topics in general topology.- 9) Intermezzo.- 10 Homotopy.- 11 The fundamental group.- 12 Covering spaces.- 13 Monodromy.- 14 van Kampen's theorem.- 15 A topological view of sheaf cohomology.- 16 Selected topics in algebraic topology.- 17 Hints and solutions.
