Topology.

Por: Dugundji, JamesSeries Allyn and Bacon series in advanced mathematicsEditor: Boston : Allyn and Bacon, 1966Descripción: xvi, 447 p. ; 24 cmTema(s): TopologyOtra clasificación: 54-01
Contenidos:
 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]
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TOPOLOGÍA


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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