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This is the softcover reprint of the English translation of 1971 (available from Springer since 1989) of the first 4 chapters of Bourbaki's Topologie générale. It gives all the basics of the subject, starting from definitions. Important classes of topological spaces are studied, uniform structures are introduced and applied to topological groups. Real numbers are constructed and their properties established. Part II, comprising the later chapters, Ch. 5-10, is also available in English in softcover.
Table des matières
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).
