Elements of Mathematics: General Topology

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.