General topology
著者
書誌事項
General topology
(Elements of mathematics / Nicolas Bourbaki)
Springer-Verlag, c1989
- ch. 1-4 : gw
- ch. 1-4 : gw : softcover
- ch. 1-4 : us
- ch. 5-10 : gw
- ch. 5-10 : gw : softcover
- ch. 5-10 : us
- タイトル別名
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Topologie générale
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熊本大学 附属図書館理(数学)
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名古屋大学 理学 図書室理数理
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Translation of: Topologie générale
Includes bibliographies and indexes
内容説明・目次
- 巻冊次
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ch. 5-10 : gw ISBN 9783540193722
内容説明
目次
- 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 -convergence.- 1. The uniformity of uniform convergence.- 2. -convergence.- 3. Examples of -convergence.- 4. Properties of the spaces ? (X
- Y).- 5. Complete subsets of ? (X: Y).- 6. -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.
- 巻冊次
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ch. 1-4 : gw ISBN 9783540193746
内容説明
- 巻冊次
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ch. 1-4 : gw : softcover ISBN 9783540642411
内容説明
目次
- of the Elements of Mathematics Series.- I. Topological Structures.- 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.- 2. Continuous functions.- 1. Continuous functions.- 2. Comparison of topologies.- 3. Initial topologies.- 4. Final topologies.- 5. Pasting together of topological spaces.- 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.- 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.- 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.- 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.- 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.- 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.- 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.- 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.- 11. 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 1.- Exercises for 2.- Exercises for 3.- Exercises for 4.- Exercises for 5.- Exercises for 6.- Exercises for 7.- Exercises for 8.- Exercises for 9.- Exercises for 10.- Exercises for 11.- Historical Note.- II. Uniform Structures.- 1. Uniform spaces.- 1. Definition of a uniform structure.- 2. Topology of a uniform space.- 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.- 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.- 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 1.- Exercises for 2.- Exercises for 3.- Exercises for 4.- Historical Note.- III: Topological Groups.- 1. Topologies on groups.- 1. Topological groups.- 2. Neighbourhoods of a point in a topological group.- 3. Isomorphisms and local isomorphisms.- 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.- 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.- 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.- 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.- 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.- 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 1.- Exercises for 2.- Exercises for 3.- Exercises for 4.- Exercises for 5.- Exercises for 6.- Exercises for 7.- Historical Note.- IV: Real Numbers.- 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.- 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.- 3. The field of real numbers.- 1. Multiplication in R.- 2. The multiplicative group R*.- 3. nth roots.- 4. The extended real line.- 1. Homeomorphism of open intervals of R.- 2. The extended line.- 3. Addition and multiplication in R?.- 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.- 6. Continuous and semi-continuous real-valued functions.- 1. Continuous real-valued functions.- 2. Semi-continuous functions.- 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.- 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 1.- Exercises for 2.- Exercises for 3.- Exercises for 4.- Exercises for 5.- Exercises for 6.- Exercises for 7.- Exercises for 8.- Historical Note.- Index of Notation (Chapters I-IV).- Index of Terminology (Chapters I-IV).
- 巻冊次
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ch. 5-10 : gw : softcover ISBN 9783540645634
内容説明
目次
- 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.
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