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
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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
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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
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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
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8.- Historical Note.- Index of Notation (Chapters I-IV).- Index of Terminology (Chapters I-IV).