InhaltsangabeV: 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 a

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One-parameter Groups.- Real Number Spaces and Projective Spaces.- The Additive Groups Rn.- Complex Numbers.- Use of Real Numbers in General Topology.- Function Spaces.- Historical Note.- Index of Notation.- Index of Terminology.