#  ## General topology  ch. 1-4 ～ ch. 5-10

Nicolas Bourbaki

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.

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This is the softcover reprint of the 1971 English translation of the first four chapters of Bourbaki's Topologie Generale. It gives all basics of the subject, starting from definitions. Important classes of topological spaces are studied, and uniform structures are introduced and applied to topological groups. In addition, real numbers are constructed and their properties established.

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[目次]

• V: One-parameter groups.- x 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.- x 2. Measurement of magnitudes.- x 3. Topological characterization of the groups R and T.- x 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 x 1.- Exercises for x 2.- Exercises for x 3.- Exercises for x 4.- Historical Note.- VI. Real number spaces and projective spaces.- x 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.- x 2. Euclidean distance, balls and spheres.- 1. Euclidean distance in Rn.- 2. Displacements.- 3. Euclidean balls and spheres.- 4. Stereographic projection.- x 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 x 1.- Exercises for x 2.- Exercises for x 3.- Historical Note.- VII. The additive groupsRn.- x 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.- x 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.- x 3. Infinite sums in the groups Rn.- 1. Summable families in Rn.- 2. Series in Rn.- Exercises for x 1.- Exercises for x 2.- Exercises for x 3.- Historical Note.- VIII. Complex numbers.- x 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.- x 2. Angular measure, trigonometric functions.- 1. The multiplicative group U.- 2. Angles.- 3. Angular measure.- 4. Trigonometric functions.- 5. Angular sectors.- 6. Crosses.- x 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.- x 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 x 1.- Exercises for x 2.- Exercises for x 3.- Exercises for x 4.- Historical Note.- IX. Use of real numbers in general topology.- x 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.- x 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.- x 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.- x 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.- x 5. Baire spaces.- 1. Nowhere dense sets.- 2. Meagre sets.- 3. Baire spaces.- 4. Semi-continuous functions on a Baire space.- x 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 x 1.- Exercises for x 2.- Exercises for x 3.- Exercises for x 4.- Exercises for x 5.- Exercises for x 6.- Exercises for the Appendix.- Historical Note.- X. Function spaces.- x1. The uniformity of -convergence.- 1. The uniformity of uniform convergence.- 2. -convergence.- 3. Examples of -convergence.- 4. Properties of the spaces S
• (X
• Y).- 5. Complete subsets of S
• (X: Y).- 6. -convergence in spaces of continuous mappings.- x 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.- x 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.- x 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 x 1.- Exercises for x 2.- Exercises for x 3.- Exercises for x 4.- Historical Note.- Index of Notation.- Index of Terminology.

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[目次]

• of the Elements of Mathematics Series.- I. Topological Structures.- x 1. Open sets, neighbourhoods, closed sets.- 1. Open sets.- 2. Neighbourhoods.- 3. Fundamental systems of neighbourhoods
• bases of a topology.- 4. Closed sets.- 5. Locally finite families.- 6. Interior, closure, frontier of a set
• dense sets.- x 2. Continuous functions.- 1. Continuous functions.- 2. Comparison of topologies.- 3. Initial topologies.- 4. Final topologies.- 5. Pasting together of topological spaces.- x 3. Subspaces, quotient spaces.- 1. Subspaces of a topological space.- 2. Continuity with respect to a subspace.- 3. Locally closed subspaces.- 4. Quotient spaces.- 5. Canonical decomposition of a continuous mapping.- 6. Quotient space of a subspace.- x 4. Product of topological spaces.- 1. Product spaces.- 2. Section of an open set
• section of a closed set, projection of an open set. Partial continuity.- 3. Closure in a product.- 4. Inverse limits of topological spaces.- x 5. Open mappings and closed mappings.- 1. Open mappings and closed mappings.- 2. Open equivalence relations and closed equivalence relations.- 3. Properties peculiar to open mappings.- 4. Properties peculiar to closed mappings.- x 6. Filters.- 1. Definition of a filter.- 2. Comparison of filters.- 3. Bases of a filter.- 4. Ultrafilters.- 5. Induced filter.- 6. Direct image and inverse image of a filter base.- 7. Product of filters.- 8. Elementary filters.- 9. Germs with respect to a filter.- 10. Germs at a point.- x 7. Limits.- 1. Limit of a filter.- 2. Cluster point of a filter base.- 3. Limit point and cluster point of a function.- 4. Limits and continuity.- 5. Limits relative to a subspace.- 6. Limits in product spaces and quotient spaces.- x 8. Hausdorff spaces and regular spaces.- 1. Hausdorff spaces.- 2. Subspaces and products of Hausdorff spaces.- 3. Hausdorff quotient spaces.- 4. Regular spaces.- 5. Extension by continuity
• double limit.- 6. Equivalence relations on a regular space.- x 9. Compact spaces and locally compact spaces.- 1. Quasi-compact spaces and compact spaces.- 2. Regularity of a compact space.- 3. Quasi-compact sets
• compact sets
• relatively compact sets.- 4. Image of a compact space under a continuous mapping.- 5. Product of compact spaces.- 6. Inverse limits of compact spaces.- 7. Locally compact spaces.- 8. Embedding of a locally compact space in a compact space.- 9. Locally compact ?-compact spaces.- 10. Paracompact spaces.- x 10. Proper mappings.- 1. Proper mappings.- 2. Characterization of proper mappings by compactness properties.- 3. Proper mappings into locally compact spaces.- 4. Quotient spaces of compact spaces and locally compact spaces.- x11. Connectedness.- 1. Connected spaces and connected sets.- 2. Image of a connected set under a continuous mapping.- 3. Quotient spaces of a connected space.- 4. Product of connected spaces.- 5. Components.- 6. Locally connected spaces.- 7. Application : the Poincare-Vol terra theorem.- Exercises for x 1.- Exercises for x 2.- Exercises for x 3.- Exercises for x 4.- Exercises for x 5.- Exercises for x 6.- Exercises for x 7.- Exercises for x 8.- Exercises for x 9.- Exercises for x 10.- Exercises for x 11.- Historical Note.- II. Uniform Structures.- x 1. Uniform spaces.- 1. Definition of a uniform structure.- 2. Topology of a uniform space.- x 2. Uniformly continuous functions.- 1. Uniformly continuous functions.- 2. Comparison of uniformities.- 3. Initial uniformities.- 4. Inverse image of a uniformity
• uniform subspaces.- 5. Least upper bound of a set of uniformities.- 6. Product of uniform spaces.- 7. Inverse limits of uniform spaces.- x 3. Complete spaces.- 1. Cauchy filters.- 2. Minimal Cauchy filters.- 3. Complete spaces.- 4. Subspaces of complete spaces.- 5. Products and inverse limits of complete spaces.- 6. Extension of uniformly continuous functions.- 7. The completion of a uniform space.- 8. The Hausdorff uniform space associated with a uniform space.- 9. Completion of subspaces and product spaces.- x 4. Relations between uniform spaces and compact spaces.- 1. Uniformity of compact spaces.- 2. Compactness of uniform spaces.- 3. Compact sets in a uniform space.- 4. Connected sets in a compact space.- Exercises for x 1.- Exercises for x 2.- Exercises for x 3.- Exercises for x 4.- Historical Note.- III: Topological Groups.- x 1. Topologies on groups.- 1. Topological groups.- 2. Neighbourhoods of a point in a topological group.- 3. Isomorphisms and local isomorphisms.- x 2. Subgroups, quotient groups, homomorphisms, homogeneous spaces, product groups.- 1. Subgroups of a topological group.- 2. Components of a topological group.- 3. Dense subgroups.- 4. Spaces with operators.- 5. Homogeneous spaces.- 6. Quotient groups.- 7. Subgroups and quotient groups of a quotient group.- 8. Continuous homomorphisms and strict morphisms.- 9. Products of topological groups.- 10. Semi-direct products.- x 3. Uniform structures on groups.- 1. The right and left uniformities on a topological group.- 2. Uniformities on subgroups, quotient groups and product groups.- 3. Complete groups.- 4. Completion of a topological group.- 5. Uniformity and completion of a commutative topological group.- x 4. Groups operating properly on a topological space
• compactness in topological groups and spaces with operators.- 1. Groups operating properly on a topological space.- 2. Properties of groups operating properly.- 3. Groups operating freely on a topological space.- 4. Locally compact groups operating properly.- 5. Groups operating continuously on a locally compact space.- 6. Locally compact homogeneous spaces.- x 5. Infinite sums in commutative groups.- 1. Summable families in a commutative group.- 2. Cauchy's criterion.- 3. Partial sums
• associativity.- 4. Summable families in a product of groups.- 5. Image of a summable family under a continuous homomorphism.- 6. Series.- 7. Commutatively convergent series.- x 6. Topological groups with operators
• topological rings, division rings and fields.- 1. Topological groups with operators.- 2. Topological direct sum of stable subgroups.- 3. Topological rings.- 4. Subrings
• ideals
• quotient rings
• products of rings.- 5. Completion of a topological ring.- 6. Topological modules.- 7. Topological division rings and fields.- 8. Uniformities on a topological division ring.- x 7. Inverse limits of topological groups and rings.- 1. Inverse limits of algebraic structures.- 2. Inverse limits of topological groups and spaces with operators.- 3. Approximation of topological groups.- 4. Application to inverse limits.- Exercises for x 1.- Exercises for x 2.- Exercises for x 3.- Exercises for x 4.- Exercises for x 5.- Exercises for x 6.- Exercises for x 7.- Historical Note.- IV: Real Numbers.- x 1. Definition of real numbers.- 1. The ordered group of rational numbers.- 2. The rational line.- 3. The real line and real numbers.- 4. Properties of intervals in R.- 5. Length of an interval.- 6. Additive uniformity of R.- x 2. Fundamental topological properties of the real line.- 1. Archimedes' axiom.- 2. Compact subsets of R.- 3. Least upper bound of a subset of R.- 4. Characterization of intervals.- 5. Connected subsets of R.- 6. Homeomorphisms of an interval onto an interval.- x 3. The field of real numbers.- 1. Multiplication in R.- 2. The multiplicative group R*.- 3. nth roots.- x 4. The extended real line.- 1. Homeomorphism of open intervals of R.- 2. The extended line.- 3. Addition and multiplication in R?.- x 5. Real-valued functions.- 1. Real-valued functions.- 2. Real-valued functions defined on a filtered set.- 3. Limits on the right and on the left of a function of a real variable.- 4. Bounds of a real-valued function.- 5. Envelopes of a family of real-valued functions.- 6. Upper limit and lower limit of a real-valued function with respect to a filter.- 7. Algebraic operations on real-valued functions.- x 6. Continuous and semi-continuous real-valued functions.- 1. Continuous real-valued functions.- 2. Semi-continuous functions.- x 7. Infinite sums and products of real numbers.- 1. Families of positive finite numbers summable in R.- 2. Families of finite numbers of arbitrary sign summable in R.- 3. Product of two infinite sums.- 4. Families multipliable in R*.- 5. Summable families and multipliable families in R.- 6. Infinite series and infinite products of real numbers.- x 8. Usual expansions of real numbers
• the power of R.- 1. Approximations to a real number.- 2. Expansions of real numbers relative to a base sequence.- 3. Definition of a real number by means of its expansion.- 4. Comparison of expansions.- 5. Expansions to base a.- 6. The power of R.- Exercises for x 1.- Exercises for x 2.- Exercises for x 3.- Exercises for x 4.- Exercises for x 5.- Exercises for x 6.- Exercises for x 7.- Exercises for x 8.- Historical Note.- Index of Notation (Chapters I-IV).- Index of Terminology (Chapters I-IV).

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### 書名 General topology Bourbaki, Nicolas Topologie générale Elements of mathematics ch. 1-4 ch. 5-10 Springer-Verlag c1989 1st ed. 1989. 2nd printing 1998 2 v. 24 cm 0387193723 038719374X 3540193723 3540642412 3540645632 354019374X BA06628659 ※クリックでCiNii Booksを表示 英語 フランス語 ドイツ

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