Elements of Mathematics. General Topology. Chapters 5-10

Inhaltsverzeichnis

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.