General Topology

of the Elements of Mathematics Series.- I. Topological Structures.- § 1. Open sets, neighbourhoods, closed sets.- § 2. Continuous functions.- § 3. Subspaces, quotient spaces.- § 4. Product of topological spaces.- § 5. Open mappings and closed mappings.- § 6. Filters.- § 7. Limits.- § 8. Hausdorff spaces and regular spaces.- § 9. Compact spaces and locally compact spaces.- § 10. Proper mappings.- §11. Connectedness.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- Exercises for § 7.- Exercises for § 8.- Exercises for § 9.- Exercises for § 10.- Exercises for § 11.- Historical Note.- II. Uniform Structures.- § 1. Uniform spaces.- § 2. Uniformly continuous functions.- § 3. Complete spaces.- § 4. Relations between uniform spaces and compact spaces.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Historical Note.- III: Topological Groups.- § 1. Topologies on groups.- § 2. Subgroups, quotient groups, homomorphisms, homogeneous spaces, product groups.- § 3. Uniform structures on groups.- § 4. Groups operating properly on a topological space; compactness in topological groups and spaces with operators.- § 5. Infinite sums in commutative groups.- § 6. Topological groups with operators; topological rings, division rings and fields.- § 7. Inverse limits of topological groups and rings.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- Exercises for § 7.- Historical Note.- IV: Real Numbers.- § 1. Definition of real numbers.- § 2. Fundamental topological properties of the real line.- § 3. The field of real numbers.- § 4. The extended real line.- § 5. Real-valued functions.- § 6. Continuous and semi-continuous real-valued functions.- § 7. Infinite sums and products of real numbers.- § 8. Usual expansions of real numbers; the power of R.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- Exercises for § 7.- Exercises for § 8.- Historical Note.- Index of Notation (Chapters I-IV).- Index of Terminology (Chapters I-IV).