General Topology: Chapters 5-10

ISBN-13: 9783540645634 / Angielski / Miękka / 1998 / 363 str.

Nicolas Bourbaki

Opis

This is the softcover reprint of the 1974 English translation of the later chapters of Bourbaki's Topologie Generale. Initial chapters study subgroups and quotients of R, real vector spaces and projective spaces, and additive groups Rn. Analogous properties are then studied for complex numbers. Later chapters illustrate the use of real numbers in general topology and discuss various topologies of function spaces and approximation of functions.

Spis treści

V: One-parameter groups.- § 1. Subgroups and quotient groups of R.- 1. Closed subgroups of R.- 2. Quotient groups of R.- 3. Continuous homomorphisms of R into itself.- 4. Local definition of a continuous homomorphism of R into a topological group.- § 2. Measurement of magnitudes.- § 3. Topological characterization of the groups R and T.- § 4. Exponentials and logarithms.- 1. Definition of ax and logax.- 2. Behaviour of the functions ax and logax.- 3. Multipliable families of numbers > 0.- 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.- 1. The topology of Rn.- 2. The additive group Rn.- 3. The vector space Rn.- 4. Affine linear varieties in Rn.- 5. Topology of vector spaces and algebras over the field R.- 6. Topology of matrix spaces over R.- § 2. Euclidean distance, balls and spheres.- 1. Euclidean distance in Rn.- 2. Displacements.- 3. Euclidean balls and spheres.- 4. Stereographic projection.- § 3. Real projective spaces.- 1. Topology of real projective spaces.- 2. Projective linear varieties.- 3. Embedding real number space in projective space.- 4. Application to the extension of real-valued functions.- 5. Spaces of projective linear varieties.- 6. Grassmannians.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Historical Note.- VII. The additive groupsRn.- § 1. Subgroups and quotient groups of Rn.- 1. Discrete subgroups of Rn.- 2. Closed subgroups of Rn.- 3. Associated subgroups.- 4. Hausdorff quotient groups of Rn.- 5. Subgroups and quotient groups of Tn.- 6. Periodic functions.- § 2. Continuous homomorphisms of Rn and its quotient groups.- 1. Continuous homomorphisms of the group Rm into the group Rn.- 2. Local definition of a continuous homomorphisms of Rn into a topological group.- 3. Continuous homomorphisms of Rm into Tn.- 4. Automorphisms of Tn.- § 3. Infinite sums in the groups Rn.- 1. Summable families in Rn.- 2. Series in Rn.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Historical Note.- VIII. Complex numbers.- § 1. Complex numbers, quaternions.- 1. Definition of complex numbers.- 2. The topology of C.- 3. The multiplicative group C*.- 4. The division ring of quaternions.- § 2. Angular measure, trigonometric functions.- 1. The multiplicative group U.- 2. Angles.- 3. Angular measure.- 4. Trigonometric functions.- 5. Angular sectors.- 6. Crosses.- § 3. Infinite sums and products of complex numbers.- 1. Infinite sums of complex numbers.- 2. Multipliable families in C*.- 3. Infinite products of complex numbers.- § 4. Complex number spaces and projective spaces.- 1. The vector space Cn.- 2. Topology of vector spaces and algebras over the field C.- 3. Complex projective spaces.- 4. Spaces of complex projective linear varieties.- 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.- 1. Pseudometrics.- 2. Definition of a uniformity by means of a family of pseudometrics.- 3. Properties of uniformities defined by families of pseudometrics.- 4. Construction of a family of pseudometrics defining a uniformity.- 5. Uniformizable spaces.- 6. Semi-continuous functions on a uniformizable space.- § 2. Metric spaces and metrizable spaces.- 1. Metrics and metric spaces.- 2. Structure of a metric space.- 3. Oscillation of a function.- 4. Metrizable uniform spaces.- 5. Metrizable topological spaces.- 6. Use of countable sequences.- 7. Semi-continuous functions on a metrizable space.- 8. Metrizable spaces of countable type.- 9. Compact metric spaces; compact metrizable spaces.- 10. Quotient spaces of metrizable spaces.- § 3. Metrizable groups, valued fields, normed spaces and algebras.- 1. Metrizable topological groups.- 2. Valued division rings.- 3. Normed spaces over a valued division ring.- 4. Quotient spaces and product spaces of normed spaces.- 5. Continuous multilinear functions.- 6. Absolutely summable families in a normed space.- 7. Normed algebras over a valued field.- § 4. Normal spaces.- 1. Definition of normal spaces.- 2. Extension of a continuous real-valued function.- 3. Locally finite open coverings of a closed set in a normal space.- 4. Paracompact spaces.- 5. Paracompactness of metrizable spaces.- § 5. Baire spaces.- 1. Nowhere dense sets.- 2. Meagre sets.- 3. Baire spaces.- 4. Semi-continuous functions on a Baire space.- § 6. Polish spaces, Souslin spaces, Borel sets.- 1. Polish spaces.- 2. Souslin spaces.- 3. Borel sets.- 4. Zero-dimensional spaces and Lusin spaces.- 5. Sieves.- 6. Separation of Souslin sets.- 7. Lusin spaces and Borel sets.- 8. Borel sections.- 9. Capacitability of Souslin sets.- Appendix: Infinite products in normed algebras.- 1. Multipliable sequences in a normed algebra.- 2. Multipliability criteria.- 3. Infinite products.- 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 𝔖-convergence.- 1. The uniformity of uniform convergence.- 2. 𝔖-convergence.- 3. Examples of 𝔖-convergence.- 4. Properties of the spaces ?𝔖 (X; Y).- 5. Complete subsets of ?𝔖 (X: Y).- 6. 𝔖-convergence in spaces of continuous mappings.- § 2. Equicontinuous sets.- 1. Definition and general criteria.- 2. Special criteria for equicontinuity.- 3. Closure of an equicontinuous set.- 4. Pointwise convergence and compact convergence on equicontinuous sets.- 5. Compact sets of continuous mappings.- § 3. Special function spaces.- 1. Spaces of mappings into a metric space.- 2. Spaces of mappings into a normed space.- 3. Countability properties of spaces of continuous functions.- 4. The compact-open topology.- 5. Topologies on groups of homeomorphisms.- § 4. Approximation of continuous real-valued functions.- 1. Approximation of continuous functions by functions belonging to a lattice.- 2. Approximation of continuous functions by polynomials.- 3. Application: approximation of continuous real-valued functions defined on a product of compact spaces.- 4. Approximation of continuous mappings of a compact space into a normed space.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Historical Note.- Index of Notation.- Index of Terminology.