Topology (Allyn and Bacon Series in Advanced Mathematics)

Front Cover......Page 1 Title Page......Page 4 Copyright Information......Page 5 Dedication......Page 6 Preface......Page 8 Contents......Page 10 Basic Notation......Page 17 1 Sets......Page 18 2 Boolean Algebra......Page 20 3 Cartesian Product......Page 24 4 Families of Sets......Page 25 6 Functions, or Maps......Page 27 7 Binary Relations; Equivalence Relations......Page 31 8 Axiomatics......Page 34 9 General Cartesian Products......Page 38 Problems......Page 42 1 Orderings......Page 46 2 Zorn's Lemma; Zermelo's Theorem......Page 48 3 Ordinals......Page 53 4 Comparability of Ordinals......Page 55 5 Transfinite Induction and Construction......Page 57 6 Ordinal Numbers......Page 58 7 Cardinals......Page 62 8 Cardinal Arithmetic......Page 66 9 The Ordinal Number Ω......Page 71 Problems......Page 74 1 Topological Spaces......Page 79 2 Basis for a Given Topology......Page 81 3 Topologizing of Sets......Page 82 4 Elementary Concepts......Page 85 5 Topologizing with Preassigned Elementary Operations......Page 89 6 G_δ, F_σ, and Borel Sets......Page 91 7 Relativization......Page 94 8 Continuous Maps......Page 95 9 Piecewise Definition of Maps......Page 98 10 Continuous Maps into E1......Page 100 11 Open Maps and Closed Maps......Page 103 12 Homeomorphism......Page 104 Problems......Page 107 1 Cartesian Product Topology......Page 115 2 Continuity of Maps......Page 118 3 Slices in Cartesian Products......Page 120 4 Peano Curves......Page 121 Problems......Page 122 1 Connectedness......Page 124 2 Applications......Page 127 3 Components......Page 128 4 Local Connectedness......Page 130 5 Path-Connectedness......Page 131 Problems......Page 133 1 Identification Topology......Page 137 2 Subspaces......Page 139 3 General Theorems......Page 140 4 Spaces with Equivalence Relations......Page 142 5 Cones and Suspensions......Page 143 6 Attaching of Spaces......Page 144 7 The Relation K(f) for Continuous Maps......Page 146 8 Weak Topologies......Page 148 Problems......Page 150 1 Hausdorff Spaces......Page 154 2 Regular Spaces......Page 158 3 Normal Spaces......Page 161 4 Urysohn's Characterization of Normality......Page 163 5 Tietze's Characterization of Normality......Page 166 6 Covering Characterization of Normality......Page 169 7 Completely Regular Spaces......Page 170 Problems......Page 173 1 Coverings of Spaces......Page 177 2 Paracompact Spaces......Page 179 3 Types of Refinements......Page 184 4 Partitions of Unity......Page 186 5 Complexes; Nerves of Coverings......Page 188 6 Second-countable Spaces; Lindelof Spaces......Page 190 7 Separability......Page 192 Problems......Page 194 1 Metrics on Sets......Page 198 2 Topology Induced by a Metric......Page 199 4 Continuity of the Distance......Page 201 5 Properties of Metric Topologies......Page 202 6 Maps of Metric Spaces into Affine Spaces......Page 204 7 Cartesian Products of Metric Spaces......Page 206 8 The Space l2(A); Hilbert Cube......Page 208 9 Metrization of Topological Spaces......Page 210 10 Gauge Spaces......Page 215 11 Uniform Spaces......Page 217 Problems......Page 221 1 Sequences and Nets......Page 226 2 Filterbases in Spaces......Page 228 3 Convergence Properties of Filterbases......Page 230 5 Continuity; Convergence in Cartesian Products......Page 232 6 Adequacy of Sequences......Page 234 7 Maximal Filterbases......Page 235 Problems......Page 237 1 Compact Spaces......Page 239 2 Special Properties of Compact Spaces......Page 243 3 Countable Compactness......Page 245 4 Compactness in Metric Spaces......Page 250 5 Perfect Maps......Page 252 6 Local Compactness......Page 254 7 σ-Compact Spaces......Page 257 8 Compactification......Page 259 9 k-Spaces......Page 264 10 Baire Spaces; Category......Page 266 Problems......Page 268 1 The Compact-open Topology......Page 274 2 Continuity of Composition; the Evaluation Map......Page 276 3 Cartesian Products......Page 277 4 Application to Identification Topologies......Page 279 5 Basis for Z^{Y}......Page 280 6 Compact Subsets of Z^{Y}......Page 282 7 Sequential Convergence in the c-Topology......Page 284 8 Metric Topologies; Relation to the c-Topology......Page 286 9 Pointwise Convergence......Page 289 10 Comparison of Topologies in Z^{Y}......Page 291 Problems......Page 292 1 Continuity of the Algebraic Operations......Page 295 2 Algebras in C^(Y;c)......Page 296 3 Stone-Weierstrass Theorem......Page 298 4 The Metric Space C(Y)......Page 301 5 Embedding of Y in C(Y)......Page 302 6 The Ring C^(Y)......Page 304 Problems......Page 307 1 Cauchy Sequences......Page 309 2 Complete Metrics and Complete Spaces......Page 310 3 Cauchy Filterbases; Total Boundedness......Page 313 4 Baire's Theorem for Complete Metric Spaces......Page 316 5 Extension of Uniformly Continuous Maps......Page 319 6 Completion of a Metric Space......Page 321 7 Fixed-Point Theorem for Complete Spaces......Page 322 8 Complete Subspaces of Complete Spaces......Page 324 9 Complete Gauge Structures......Page 325 Problems......Page 328 1 Homotopy......Page 332 2 Homotopy Classes......Page 334 3 Homotopy and Function Spaces......Page 336 4 Relative Homotopy......Page 338 5 Retracts and Extendability......Page 339 6 Deformation Retraction and Homotopy......Page 340 7 Homotopy and Extendability......Page 343 8 Applications......Page 347 Problems......Page 349 1 Degree of a Map Sn → Sn......Page 352 2 Brouwer's Theorem......Page 357 3 Further Applications of the Degree of a Map......Page 358 4 Maps of Spheres into Sn......Page 360 5 Maps of Spaces into Sn......Page 363 6 Borsuk's Antipodal Theorem......Page 364 7 Degree and Homotopy......Page 367 Problems......Page 370 XVII. Topology of En......Page 372 1 Components of Compact Sets in En+1......Page 373 2 Borsuk's Separation Theorem......Page 374 3 Domain Invariance......Page 375 4 Deformations of Subsets of En+1......Page 376 5 The Jordan Curve Theorem......Page 378 Problems......Page 380 1 Homotopy Type......Page 382 2 Homotopy-Type Invariants......Page 384 4 Mapping Cylinder......Page 385 5 Properties of X in C(f)......Page 388 6 Change of Bases in C(f)......Page 389 Problems......Page 391 1 Path Spaces......Page 393 2 H-Structures......Page 396 3 H-Homomorphisms......Page 398 4 H-Spaces......Page 400 5 Units......Page 401 6 Inversion......Page 403 7 Associativity......Page 404 8 Path Spaces on H-Spaces......Page 405 Problems......Page 407 1 Fiber Spaces......Page 409 2 Fiber Spaces for the Class of All Spaces......Page 412 3 The Uniformization Theorem of Hurewicz......Page 416 4 Locally Trivial Fiber Structures......Page 421 Problems......Page 425 Appendix One: Vector Spaces; Polytopes......Page 427 Appendix Two: Direct and Inverse Limits......Page 437 Index......Page 454