Topology

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 Zorns lemma topological spaces and continuous functions connectedness and compactness Alexandrov compactification quotient topologies countability and separation axioms prebasis and Alexanders 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 Borsuks 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. It is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.

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.- Homotopy.- 10 The fundamental group.- 11 Covering spaces.- Monodromy.- 12 van Kampens theorem.- 13 Selected topics in algebraic topology.- 14 Hints and solutions.- 15 References.- 16 Index.