Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.
Ce deuxième volume du Livre de Topologie générale, troisième Livre du traité, décrit de nombreux outils fondamentaux en topologie et en analyse, tels que le théorème d Urysohn, le théorème de Baire ou les espaces polonais. Il comprend les chapitres: 1. Groupes à un paramètre; 2. Espaces numériques et espaces projectifs; 3. Les groupes additifs Rn; 4. Nombres complexes; 5. Utilisation des nombres réels en topologie générale; 6. Espaces fonctionnels.
Il contient également des notes historiques.
Inhaltsverzeichnis
V: One-parameter groups. - § 1. Subgroups and quotient groups of R. - § 2. Measurement of magnitudes. - § 3. Topological characterization of the groups R and T. - § 4. Exponentials and logarithms. - Exercises for § 1. - Exercises for § 2. - Exercises for § 3. - Exercises for § 4. - Historical Note. - VI. Real number spaces and projective spaces. - § 1. Real number space Rn. - § 2. Euclidean distance, balls and spheres. - § 3. Real projective spaces. - Exercises for § 1. - Exercises for § 2. - Exercises for § 3. - Historical Note. - VII. The additive groupsRn. - § 1. Subgroups and quotient groups of Rn. - § 2. Continuous homomorphisms of Rn and its quotient groups. - § 3. Infinite sums in the groups Rn. - Exercises for § 1. - Exercises for § 2. - Exercises for § 3. - Historical Note. - VIII. Complex numbers. - § 1. Complex numbers, quaternions. - § 2. Angular measure, trigonometric functions. - § 3. Infinite sums and products of complex numbers. - § 4. Complex number spaces and projective spaces. - Exercises for § 1. - Exercises for § 2. - Exercises for § 3. - Exercises for § 4. - Historical Note. - IX. Use of real numbers in general topology. - § 1. Generation of a uniformity by a family of pseudometrics; uniformizable spaces. - § 2. Metric spaces and metrizable spaces. - § 3. Metrizable groups, valued fields, normed spaces and algebras. - § 4. Normal spaces. - § 5. Baire spaces. - § 6. Polish spaces, Souslin spaces, Borel sets. - Appendix: Infinite products in normed algebras. - Exercises for § 1. - Exercises for § 2. - Exercises for § 3. - Exercises for § 4. - Exercises for § 5. - Exercises for § 6. - Exercises for the Appendix. - Historical Note. - X. Function spaces. - §1. The uniformity of -convergence. - § 2. Equicontinuous sets. - § 3. Special function spaces. - § 4. Approximation of continuous real-valued functions. - Exercises for § 1. - Exercises for § 2. - Exercises for § 3. - Exercises for § 4. - Historical Note. - Index of Notation. - Index of Terminology.
