Topology

Main subject categories: • Point-set topology • Manifolds • Homotopy • The fundamental group • Covering spaces • Sheaf cohomology

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; connectedness and compactness; Alexandrov compactification; quotient topologies; 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.

Provides a comprehensive treatment of the basic theory and of some complementary topics. • Contains lots of exercises, of various difficulty level • Contains an original topological approach to sheaf cohomology