Topology
Algebraic topology Mathematics Mathematics, general
Imprint: Springer
2015
1st ed. 2015.
EISBN 3319169580
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 Kampen's theorem.
13 Selected topics in algebraic topology.
14 Hints and solutions.
15 References.
16 Index.
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. It is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.
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 Kampen's theorem.
13 Selected topics in algebraic topology.
14 Hints and solutions.
15 References.
16 Index.
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. It is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.
