Topology.
Series Allyn and Bacon series in advanced mathematicsEditor: Boston : Allyn and Bacon, 1966Descripción: xvi, 447 p. ; 24 cmTema(s): TopologyOtra clasificación: 54-01Contents I. Elementary Set Theory I 1 Sets [1] 2 Boolean Algebra [3] 3 Cartesian Product [7] 4 Families of Sets [8] 5 Power Set [10] 6 Functions, or Maps [10] 7 Binary Relations; Equivalence Relations [14] 8 Axiomatics [17] 9 General Cartesian Products [21] Problems [25] Ordinals and Cardinals [29] 1 Orderings [29] 2 Zorn’s Lemma; Zermelo’s Theorem [31] 3 Ordinals [36] 4 Comparability of Ordinals [38] 5 Transfinite Induction and Construction [40] 6 Ordinal Numbers [41] 7 Cardinals [45] 8 Cardinal Arithmetic [49] 9 The Ordinal Number Ω [54] Problems [57] III. Topological Spaces [62] 1 Topological Spaces [62] 2 Basis for a Given Topology [64] 3 Topologizing of Sets [65] 4 Elementary Concepts [68] 5 Topologizing with Preassigned Elementary Operations [72] 6 G6i Fai and Borel Sets [74] 7 Relativization [77] 8 Continuous Maps [78] 9 Piecewise Definition of Maps [81] 10 Continuous Maps into E1 [83] 11 Open Maps and Closed Maps [86] 12 Homeomorphism [87] Problems [90] IV. Cartesian Products [98] 1 Cartesian Product Topology [98] 2 Continuity of Maps [101] 3 Slices in Cartesian Products [103] 4 Peano Curves [104] Problems [105] V. Connectedness [107] 1 Connectedness [107] 2 Applications [110] 3 Components [111] 4 Local Connectedness [113] 5 Path-Connectedness [114] Problems [116] VI. Identification Topology; Weak Topology [120] 1 Identification Topology [120] 2 Subspaces [122] 3 General Theorems [123] 4 Spaces with Equivalence Relations [125] 5 Cones and Suspensions [126] 6 Attaching of Spaces [127] 7 The Relation K(f) for Continuous Maps [129] 8 Weak Topologies [131] Problems [133] VII. Separation Axioms [137] 1 Hausdorff Spaces [137] 2 Regular Spaces [141] 3 Normal Spaces [144] 4 Urysohn’s Characterization of Normality [146] 5 Tietze’s Characterization of Normality [149] 6 Covering Characterization of Normality [152] 7 Completely Regular Spaces [153] Problems [156] VIII. Covering Axioms [160] 1 Coverings of Spaces [160] 2 Paracompact Spaces [162] 3 Types of Refinements [167] 4 Partitions of Unity [169] 5 Complexes; Nerves of Coverings [171] 6 Second-countable Spaces; Lindelöf Spaces [173] 7 Separability [175] Problems [177] IX. Metric Spaces [181] 1 Metrics on Sets [181] 2 Topology Induced by a Metric [182] 3 Equivalent Metrics [184] 4 Continuity of the Distance [184] 5 Properties of Metric Topologies [185] 6 Maps of Metric Spaces into Affine Spaces [187] 7 Cartesian Products of Metric Spaces [189] 8 The Space l2(A); Hilbert Cube [191] 9 Metrization of Topological Spaces [193] 10 Gauge Spaces [198] 11 Uniform Spaces [200] Problems [204] X. Convergence [209] 1 Sequences and Nets [209] 2 Filterbases in Spaces [211] 3 Convergence Properties of Filterbases [213] 4 Closure in Terms of Filterbases [215] 5 Continuity; Convergence in Cartesian Products [215] 6 Adequacy of Sequences [217] 7 Maximal Filterbases [218] Problems [220] XI. Compactness [222] 1 Compact Spaces [222] 2 Special Properties of Compact Spaces [226] 3 Countable Compactness [228] 4 Compactness in Metric Spaces [233] 5 Perfect Maps [235] 6 Local Compactness [237] 7 σ-Compact Spaces [240] 8 Compactification [242] 9 k-Spaces [247] 10 Baire Spaces; Category [249] Problems [251] XII. Function Spaces [257] 1 The Compact-open Topology [257] 2 Continuity of Composition; the Evaluation Map [259] 3 Cartesian Products [260] 4 Application to Identification Topologies [262] 5 Basis for ZY [263] 6 Compact Subsets of ZY [265] 7 Sequential Convergence in the ^-Topology [267] 8 Metric Topologies; Relation to the c-Topology [269] 9 Pointwise Convergence [272] 10 Comparison of Topologies in ZY [274] Problems [275] XIII. The Spaces C(Y) [278] 1 Continuity of the Algebraic Operations [278] 2 Algebras in Ĉ(Y; c) [219] 3 Stone-Weierstrass Theorem [281] 4 The Metric Space C(Y) [284] 5 Embedding of Y in C( Y) [285] 6 The Ring Ĉ(Y) [287] Problems [290] XIV. Complete Spaces [292] 1 Cauchy Sequences [292] 2 Complete Metrics and Complete Spaces [293] 3 Cauchy Filterbases; Total Boundedness [296] 4 Baire’s Theorem for Complete Metric Spaces [299] 5 Extension of Uniformly Continuous Maps [302] 6 Completion of a Metric Space [304] 7 Fixed-Point Theorem for Complete Spaces [305] 8 Complete Subspaces of Complete Spaces [307] 9 Complete Gauge Structures [309] Problems [312] XV. Homotopy [315] 1 Homotopy [315] 2 Homotopy Classes [317] 3 Homotopy and Function Spaces [319] 4 Relative Homotopy [321] 5 Retracts and Extendability [322] 6 Deformation Retraction and Homotopy [323] 7 Homotopy and Extendability [326] 8 Applications [330] Problems [332] XVI. Maps into Spheres [335] 1 Degree of a Map Sn -> Sn [335] 2 Brouwer’s Theorem [340] 3 Further Applications of the Degree of a Map [341] 4 Maps of Spheres into Sn [343] 5 Maps of Spaces into Sn [346] 6 Borsuk’s Antipodal Theorem [347] 7 Degree and Homotopy [350] Problems [353] XVII. Topology of En [355] 1 Components of Compact Sets in En+1 [356] 2 Borsuk’s Separation Theorem [357] 3 Domain Invariance [358] 4 Deformations of Subsets of En+1 [359] 5 The Jordan Curve Theorem [361] Problems [363] XVIII. Homotopy Type [365] 1 Homotopy Type [365] 2 Homotopy-Type Invariants [367] 3 Homotopy of Pairs [368] 4 Mapping Cylinder [368] 5 Properties of X in C(f) [371] 6 Change of Bases in C(f) [372] Problems [374] XIX. Path Spaces; H-Spaces [376] 1 Path Spaces [376] 2 H- Structures [379] 3 H-Homomorphisms [381] 4 H-Spaces [383] 5 Units [384] 6 Inversion [386] 7 Associativity [387] 8 Path Spaces on H-Spaces [388] Problems [390] XX. Fiber Spaces [392] 1 Fiber Spaces [392] 2 Fiber Spaces for the Class of All Spaces [395] 3 The Uniformization Theorem of Hurewicz [399] 4 Locally Trivial Fiber Structures [404] Problems [408] Appendix One: Vector Spaces; Polytopes [410] Appendix Two: Direct and Inverse Limits [420] Index [437]
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 54 D868 (Browse shelf) | Available | A-2251 | ||||
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| 54 C748 Introduzione alla topologia / | 54 C958 Fondements de la topologie générale / | 54 C958i Foundations of general topology / | 54 D868 Topology. | 54 D868 Topology. | 54 En57 General topology / | 54 F272 Espace et dimension / |
Contents --
I. Elementary Set Theory I --
1 Sets [1] --
2 Boolean Algebra [3] --
3 Cartesian Product [7] --
4 Families of Sets [8] --
5 Power Set [10] --
6 Functions, or Maps [10] --
7 Binary Relations; Equivalence Relations [14] --
8 Axiomatics [17] --
9 General Cartesian Products [21] --
Problems [25] --
Ordinals and Cardinals [29] --
1 Orderings [29] --
2 Zorn’s Lemma; Zermelo’s Theorem [31] --
3 Ordinals [36] --
4 Comparability of Ordinals [38] --
5 Transfinite Induction and Construction [40] --
6 Ordinal Numbers [41] --
7 Cardinals [45] --
8 Cardinal Arithmetic [49] --
9 The Ordinal Number Ω [54] --
Problems [57] --
III. Topological Spaces [62] --
1 Topological Spaces [62] --
2 Basis for a Given Topology [64] --
3 Topologizing of Sets [65] --
4 Elementary Concepts [68] --
5 Topologizing with Preassigned Elementary Operations [72] --
6 G6i Fai and Borel Sets [74] --
7 Relativization [77] --
8 Continuous Maps [78] --
9 Piecewise Definition of Maps [81] --
10 Continuous Maps into E1 [83] --
11 Open Maps and Closed Maps [86] --
12 Homeomorphism [87] --
Problems [90] --
IV. Cartesian Products [98] --
1 Cartesian Product Topology [98] --
2 Continuity of Maps [101] --
3 Slices in Cartesian Products [103] --
4 Peano Curves [104] --
Problems [105] --
V. Connectedness [107] --
1 Connectedness [107] --
2 Applications [110] --
3 Components [111] --
4 Local Connectedness [113] --
5 Path-Connectedness [114] --
Problems [116] --
VI. Identification Topology; Weak Topology [120] --
1 Identification Topology [120] --
2 Subspaces [122] --
3 General Theorems [123] --
4 Spaces with Equivalence Relations [125] --
5 Cones and Suspensions [126] --
6 Attaching of Spaces [127] --
7 The Relation K(f) for Continuous Maps [129] --
8 Weak Topologies [131] --
Problems [133] --
VII. Separation Axioms [137] --
1 Hausdorff Spaces [137] --
2 Regular Spaces [141] --
3 Normal Spaces [144] --
4 Urysohn’s Characterization of Normality [146] --
5 Tietze’s Characterization of Normality [149] --
6 Covering Characterization of Normality [152] --
7 Completely Regular Spaces [153] --
Problems [156] --
VIII. Covering Axioms [160] --
1 Coverings of Spaces [160] --
2 Paracompact Spaces [162] --
3 Types of Refinements [167] --
4 Partitions of Unity [169] --
5 Complexes; Nerves of Coverings [171] --
6 Second-countable Spaces; Lindelöf Spaces [173] --
7 Separability [175] --
Problems [177] --
IX. Metric Spaces [181] --
1 Metrics on Sets [181] --
2 Topology Induced by a Metric [182] --
3 Equivalent Metrics [184] --
4 Continuity of the Distance [184] --
5 Properties of Metric Topologies [185] --
6 Maps of Metric Spaces into Affine Spaces [187] --
7 Cartesian Products of Metric Spaces [189] --
8 The Space l2(A); Hilbert Cube [191] --
9 Metrization of Topological Spaces [193] --
10 Gauge Spaces [198] --
11 Uniform Spaces [200] --
Problems [204] --
X. Convergence [209] --
1 Sequences and Nets [209] --
2 Filterbases in Spaces [211] --
3 Convergence Properties of Filterbases [213] --
4 Closure in Terms of Filterbases [215] --
5 Continuity; Convergence in Cartesian Products [215] --
6 Adequacy of Sequences [217] --
7 Maximal Filterbases [218] --
Problems [220] --
XI. Compactness [222] --
1 Compact Spaces [222] --
2 Special Properties of Compact Spaces [226] --
3 Countable Compactness [228] --
4 Compactness in Metric Spaces [233] --
5 Perfect Maps [235] --
6 Local Compactness [237] --
7 σ-Compact Spaces [240] --
8 Compactification [242] --
9 k-Spaces [247] --
10 Baire Spaces; Category [249] --
Problems [251] --
XII. Function Spaces [257] --
1 The Compact-open Topology [257] --
2 Continuity of Composition; the Evaluation Map [259] --
3 Cartesian Products [260] --
4 Application to Identification Topologies [262] --
5 Basis for ZY [263] --
6 Compact Subsets of ZY [265] --
7 Sequential Convergence in the ^-Topology [267] --
8 Metric Topologies; Relation to the c-Topology [269] --
9 Pointwise Convergence [272] --
10 Comparison of Topologies in ZY [274] --
Problems [275] --
XIII. The Spaces C(Y) [278] --
1 Continuity of the Algebraic Operations [278] --
2 Algebras in Ĉ(Y; c) [219] --
3 Stone-Weierstrass Theorem [281] --
4 The Metric Space C(Y) [284] --
5 Embedding of Y in C( Y) [285] --
6 The Ring Ĉ(Y) [287] --
Problems [290] --
XIV. Complete Spaces [292] --
1 Cauchy Sequences [292] --
2 Complete Metrics and Complete Spaces [293] --
3 Cauchy Filterbases; Total Boundedness [296] --
4 Baire’s Theorem for Complete Metric Spaces [299] --
5 Extension of Uniformly Continuous Maps [302] --
6 Completion of a Metric Space [304] --
7 Fixed-Point Theorem for Complete Spaces [305] --
8 Complete Subspaces of Complete Spaces [307] --
9 Complete Gauge Structures [309] --
Problems [312] --
XV. Homotopy [315] --
1 Homotopy [315] --
2 Homotopy Classes [317] --
3 Homotopy and Function Spaces [319] --
4 Relative Homotopy [321] --
5 Retracts and Extendability [322] --
6 Deformation Retraction and Homotopy [323] --
7 Homotopy and Extendability [326] --
8 Applications [330] --
Problems [332] --
XVI. Maps into Spheres [335] --
1 Degree of a Map Sn -> Sn [335] --
2 Brouwer’s Theorem [340] --
3 Further Applications of the Degree of a Map [341] --
4 Maps of Spheres into Sn [343] --
5 Maps of Spaces into Sn [346] --
6 Borsuk’s Antipodal Theorem [347] --
7 Degree and Homotopy [350] --
Problems [353] --
XVII. Topology of En [355] --
1 Components of Compact Sets in En+1 [356] --
2 Borsuk’s Separation Theorem [357] --
3 Domain Invariance [358] --
4 Deformations of Subsets of En+1 [359] --
5 The Jordan Curve Theorem [361] --
Problems [363] --
XVIII. Homotopy Type [365] --
1 Homotopy Type [365] --
2 Homotopy-Type Invariants [367] --
3 Homotopy of Pairs [368] --
4 Mapping Cylinder [368] --
5 Properties of X in C(f) [371] --
6 Change of Bases in C(f) [372] --
Problems [374] --
XIX. Path Spaces; H-Spaces [376] --
1 Path Spaces [376] --
2 H- Structures [379] --
3 H-Homomorphisms [381] --
4 H-Spaces [383] --
5 Units [384] --
6 Inversion [386] --
7 Associativity [387] --
8 Path Spaces on H-Spaces [388] --
Problems [390] --
XX. Fiber Spaces [392] --
1 Fiber Spaces [392] --
2 Fiber Spaces for the Class of All Spaces [395] --
3 The Uniformization Theorem of Hurewicz [399] --
4 Locally Trivial Fiber Structures [404] --
Problems [408] --
Appendix One: Vector Spaces; Polytopes [410] --
Appendix Two: Direct and Inverse Limits [420] --
Index [437] --
MR, 57 #17581
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