Munkres J. — Topology :: Электронная библиотека попечительского совета мехмата МГУ

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Munkres J. — Topology

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Название: Topology

Автор: Munkres J.

Аннотация:

This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. Includes many examples and figures. GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness and Compactness. Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness. Complete Metric Spaces and Function Spaces. Baire Spaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The Fundamental Group. Separation Theorems. The Seifert-van Kampen Theorem. Classification of Surfaces. Classification of Covering Spaces. Applications to Group Theory. For anyone needing a basic, thorough, introduction to general and algebraic topology and its applications.

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Рубрика: Математика/Геометрия и топология/Общая топология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-e издание

Год издания: 2000

Количество страниц: 276

Добавлена в каталог: 20.04.2005

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Предметный указатель

$B(x, \epsilon)$       119
$B^{2}$       135 (see also “ $B^{n}$ ”)
$B^{n}$       156
$B^{n}$ , compactness      174
$B^{n}$ , fundamental group      331
$B^{n}$ , path connectedness      156
$e_{x}$ (constant path)      327
$F_{\sigma}$ set      252
$G_{\delta}$ set      194 249
$h_{*}$       333
$h_{*}$ , dependence on base point      335
$h_{*}$ , functorial properties      334
$H_{1}(X)$       (see “First homology group”)
$I^{2}_{o}$       (see “Ordered square”)
$l^{2}$ -topology      128
$P^{2}$       372
$P^{2}$ is surface      372
$P^{2}$ , fundamental group      373
$Q^{\infty}$       195
$S^{1}$       106
$S^{1}$ , covering spaces      337 482
$S^{1}$ , fundamental group      345
$S^{2}$       139
$S^{2}$ as quotient space      136 139
$S^{n}$ (unit sphere)      156
$S^{n}$ (unit sphere), compactness      174
$S^{n}$ (unit sphere), fundamental group      369
$S^{n}$ (unit sphere), path connectedness      156
$S^{n}$ (unit sphere), simple connectedness      369
$S_{\alpha}$ (section of well-ordered set)      66
$S_{\Omega}$       66
$S_{\Omega}$ , compactification      242
$S_{\Omega}$ , countable subsets      66 74
$S_{\Omega}$ , existence      74
$S_{\Omega}$ , metrizability      181
$S_{\Omega}$ , paracompactness      260 261
$S_{\Omega}$ , uniqueness      73
$S_{\Omega}\times\bar{S_{\Omega}}$       203
$S_{\Omega}\times\bar{S_{\Omega}}$ , complete regularity      212
$S_{\Omega}\times\bar{S_{\Omega}}$ , normality      203
$S_{\Omega}\times\bar{S_{\Omega}}$ , paracompactness      254
$T_{1}$ axiom      99
$T_{1}$ axiom for quotient space      141
$T_{1}$ axiom vs. Hausdorff condition      99
$T_{1}$ axiom vs. limit points      99
$T_{i}$ axioms      211 213
$U(A, \epsilon)$       111
$X^{J}$       113
$X^{m}$       38
$X^{\omega}$       38
$\bar{A}$ (closure)      95
$\bar{D}$       121
$\bar{S_{\Omega}}$       66
$\bar{S_{\Omega}}$ , metrizability      181
$\bar{\rho}$       124 266
$\beta(x)$       (see “Stone — $\check{C}$ ech compactification”)
$\hat{\alpha}$       331
$\hat{\alpha}$ independence of path      335
$\hat{\alpha}$ is isomorphism      332
$\mathbb{R}$ (reals)      30
$\mathbb{R}$ (reals), algebraic properties      30
$\mathbb{R}$ (reals), compact subspaces      173
$\mathbb{R}$ (reals), connected subspaces      154
$\mathbb{R}$ (reals), K-topology      82 (see also “ $\mathbb{R}_{K}$ ”)
$\mathbb{R}$ (reals), local compactness      182
$\mathbb{R}$ (reals), lower limit topology      82 (see also “ $\mathbb{R}_{l}$ ”)
$\mathbb{R}$ (reals), metric for      120
$\mathbb{R}$ (reals), order properties      31
$\mathbb{R}$ (reals), second-countability      190
$\mathbb{R}$ (reals), standard topology      81
$\mathbb{R}$ (reals), uncountability      177
$\mathbb{R}^{2}$ , standard topology      87
$\mathbb{R}^{2}-0$       325
$\mathbb{R}^{2}-0$ , covering space      340
$\mathbb{R}^{2}-0$ , fundamental group      360
$\mathbb{R}^{2}_{l}$       193
$\mathbb{R}^{2}_{l}$ , complete regularity      212
$\mathbb{R}^{2}_{l}$ , Lindeloef condition      193
$\mathbb{R}^{2}_{l}$ , paracompactness      257
$\mathbb{R}^{2}_{l}$ , separation axioms      198
$\mathbb{R}^{J}$ in box topology is Baire      300
$\mathbb{R}^{J}$ in box topology, complete regularity      213
$\mathbb{R}^{J}$ in product topology (cont.) normality      203
$\mathbb{R}^{J}$ in product topology (cont.) paracompactness      257
$\mathbb{R}^{J}$ in product topology countable dense subset      195
$\mathbb{R}^{J}$ in product topology is Baire      300
$\mathbb{R}^{J}$ in product topology metrizability      133
$\mathbb{R}^{J}$ in uniform topology      124
$\mathbb{R}^{J}$ in uniform topology completeness      267
$\mathbb{R}^{J}$ in uniform topology is Baire      300
$\mathbb{R}^{l}$ , countable dense subset      195
$\mathbb{R}^{n} - 0$       156
$\mathbb{R}^{n} - 0$ , fundamental group      360
$\mathbb{R}^{n} - 0$ , path connectedness      156
$\mathbb{R}^{n}$       38
$\mathbb{R}^{n}$ , basis      116
$\mathbb{R}^{n}$ , compact subspaces      173
$\mathbb{R}^{n}$ , local compactness      182
$\mathbb{R}^{n}$ , metrics for      122 123
$\mathbb{R}^{n}$ , paracompactness      253
$\mathbb{R}^{n}$ , second-countability      190
$\mathbb{R}^{\infty}$       118
$\mathbb{R}^{\infty}$ , closure in $\mathbb{R}^{\omega}$       118 127
$\mathbb{R}^{\infty}$ , paracompactness      260
$\mathbb{R}^{\omega}$       38
$\mathbb{R}^{\omega}$ in box topology, components      162
$\mathbb{R}^{\omega}$ in box topology, connectedness      151
$\mathbb{R}^{\omega}$ in box topology, metrizability      132
$\mathbb{R}^{\omega}$ in box topology, normality      205
$\mathbb{R}^{\omega}$ in box topology, paracompactness      205
$\mathbb{R}^{\omega}$ in product topology, completeness      265
$\mathbb{R}^{\omega}$ in product topology, connectedness      151
$\mathbb{R}^{\omega}$ in product topology, local compactness      182
$\mathbb{R}^{\omega}$ in product topology, metrizability      125
$\mathbb{R}^{\omega}$ in product topology, paracompactness      257
$\mathbb{R}^{\omega}$ in product topology, second-countability      190
$\mathbb{R}^{\omega}$ in uniform topology, components      162
$\mathbb{R}^{\omega}$ in uniform topology, paracompactness      257
$\mathbb{R}^{\omega}$ in uniform topology, second-countability      190
$\mathbb{R}_{+}$       31
$\mathbb{R}_{K}$       82
$\mathbb{R}_{K}$ , connectedness      178
$\mathbb{R}_{K}$ , separation axioms      197
$\mathbb{R}_{K}$ , vs. standard topology      82
$\mathbb{R}_{l}$       82
$\mathbb{R}_{l}$ , countability axioms      192
$\mathbb{R}_{l}$ , metrizability      194
$\mathbb{R}_{l}$ , normality      198
$\mathbb{R}_{l}$ , paracompactness      257
$\mathbb{R}_{l}$ , vs. standard topology      82
$\mathbb{Z}$       32
$\mathbb{Z}_{+}$       32
$\mathbb{Z}_{+}$ , not finite      42
$\mathbb{Z}_{+}$ , well-ordered      32
$\mathcal{C}(E, p, B)$       487 (see also “Group of covering transformations”)
$\mathfrak{P}(A)$       12
$\omega$ -tuple      38
$\pi(X, x_{0})$       331 (see also “Fundamental group”)
$\rho$       122 268
$\sigma$ -compact      289 316
$\sigma$ -locally discrete      252
$\sigma$ -locally finite      245
2-cell      441
2-manifold      225
2-manifold with boundary      476
2-manifold, topological dimension      308 352
2-sphere      139 (see also “ $S^{2}$ ”)
Absolute retract      223
Absolute retract vs. universal extension property      223 224
Accumulation point of a net      188

Action of a group on a space      199
Adjoining a 2-cell      441
Adjoining a 2-cell, effect on fundamental group      439
Adjunction space      224
Affinely independent      309
Alexander homed sphere      393
Algebraic number      51
Antipodal map      372
Antipode      356
Antipode-preserving      356
ARC      308 379
Arc does not separate $S^{2}$       389
Archimedean ordering      33
Arzela’s theorem      280 293
Ascoli’s Theorem      278 290
Axiom of Choice      59
Axiom of choice vs. nonemptiness of product      118
Axiom of choice vs. well-ordering theorem      73
Axiom of choice, finite      61
Baire category theorem      296
Baire category theorem, special case      178
Baire space      295
Baire space (cont.), $\mathbb{R}^{J}$ in box, product, uniform topologies      300
Baire space (cont.), open subspace of Baire space      297
Baire space, compact Hausdorff space      296
Baire space, complete metric space      296
Baire space, fine topology on $\mathcal{C}(X, Y)$       300
Baire space, locally compact Hausdorff space      299
Ball, unit      135 156
Barber of Seville paradox      47
Base point      331
Base point choice, effect on $h_{*}$       335
Base point choice, effect on $\pi_{1}$       332
Basis for a free abelian group      411
Basis for a topology      78 80
Basis for box topology      115 116
Bd A      102
Betti number      424
Bicompactness      178
Bijective function      18
Binary operation      30
Bing metrization theorem      252
Bisection theorem      358 359
Borsuk lemma      382 385
Borsuk — Ulam theorem      358 359
Boundary of a set      102
Boundary of a surface with boundary      476
Bounded above      27
Bounded below      27
Bounded function      267
Bounded metric      121 129
Bounded set      121
Box topology      114
Box topology vs. fine topology      290
Box topology vs. product topology      115
Box topology vs. uniform topology      124 289 290
Box topology, Hausdorff condition      116
Box topology, subspace      116
Brouwer fixed-point theorem      351 353
Cantor set      178
Cardinality of a finite set      39 42
Cardinality, comparability      68
Cardinality, greater      62
Cardinality, same      51
Cartesian product, countably infinite      38
Cartesian product, finite      13 37
Cartesian product, general      113
Cauchy integral formula      405
Cauchy sequence      264
Choice axiom      (see “Axiom of choice”)
Choice function      59
Circle, unit      (see “ $S^{1}$ ”)
Classification of covering spaces      482
Classification of covering transformations      488
Classification of surfaces      469
Clockwise loop      405
Closed edge path      566
Closed graph theorem      171
Closed interval      84
Closed map      137
Closed ray      86
Closed refinement      245
Closed set      93
Closed set in subspace      94
Closed set vs. limit points      98
Closed topologist’s sine curve      381
Closed topologist’s sine curve, separates $S^{2}$       381 393
Closure      95
Closure (cont.) via limit points      97
Closure (cont.) via sequences      130 190
Closure in a cartesian product      101 116
Closure in a subspace      95
Closure of a connected subspace      150
Closure of a union      101 245
Closure via basis elements      96
Closure via nets      187
Coarser topology      77
Cofinal      187
Coherent topology      224 435
Collection      12
Commutator      422
Commutator subgroup      422
Compact      164 (see also “Compact Hausdorff space”; “Compactness”)
Compact convergence topology      283
Compact convergence topology vs. compact-open topology      285
Compact convergence topology vs. pointwise convergence topology      285
Compact convergence topology vs. uniform topology      285
Compact convergence topology, independence of metric      286
Compact Hausdorff space is Baire space      296
Compact Hausdorff space, components equal quasicomponents      236
Compact Hausdorff space, metrizability      218
Compact Hausdorff space, normality      202
Compact Hausdorff space, paracompactness      252
Compact-open topology      285
Compact-open topology continuity of evaluation map      286
Compact-open topology vs. compact convergence topology      285
Compactification      185 237
Compactification of (0, 1)      238
Compactification, induced by an imbedding      238
Compactification, one-point      185
Compactly generated space      283
Compactness      164 (see also “Compact Hausdorff space”)
Compactness and least upper bound property      172
Compactness and perfect maps      172
Compactness closed set criterion      169
Compactness in $\mathbb{R}$ and $\mathbb{R}^{n}$       173
Compactness in $\mathcal{C}(X, Y)$       290 293
Compactness in $\mathcal{C}(X, \mathbb{R}^{n})$       278 279
Compactness in finite complement topology      166
Compactness in Hausdorff metric      281
Compactness in order topology      172
Compactness of closed intervals in M      173
Compactness of continuous image      166
Compactness of countable products      280
Compactness of finite products      167
Compactness of product space      234 236
Compactness of subspace      164
Compactness via nets      188
Compactness vs. completeness      276
Compactness vs. limit point compactness      179
Compactness vs. sequential compactness      179
Comparability of cardinalities      68
Comparability of topologies      77
Comparability of well-ordered sets      73
Comparison test for infinite series      135
Complement      10
Complete graph      394
Complete graph on five vertices      308 397
Complete metric space      264 (see also “Completeness”)
Complete regularity      211

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