Topics on Continua

This book is a significant companion text to the existing literature on continuum theory. It opens with background information of continuum theory, so often missing from the preceding publications, and then explores the following topics: inverse limits, the Jones set function T, homogenous continua, and n-fold hyperspaces. In this new edition of the book, the author builds on the aforementioned topics, including the unprecedented presentation of n-fold hyperspace suspensions and induced maps on n-fold hyperspaces. The first edition of the book has had a remarkable impact on the continuum theory community. After twelve years, this updated version will also prove to be an excellent resource within the field of topology.-- Provided by publisher Preface to the Second Edition 7 Preface to the First Edition 10 Contents 13 1 Preliminaries 16 1.1 Product Topology 16 1.2 Continuous Decompositions 20 1.3 Homotopy and Fundamental Group 31 1.4 Geometric Complexes and Polyhedra 40 1.5 Complete Metric Spaces 43 1.6 Compacta 45 1.7 Continua 48 1.8 Hyperspaces 59 References 65 2 Inverse Limits and Related Topics 67 2.1 Inverse Limits 67 2.2 Inverse Limits and the Cantor Set 85 2.3 Inverse Limits and Other Operations 91 2.4 Chainable Continua 96 2.5 Circularly Chainable and P-Like Continua 107 2.6 Universal and AH-Essential Maps 114 References 124 3 Jones's Set Function T 126 3.1 The Set Function T 126 3.2 Idempotency of T 154 3.3 Continuity of T 160 3.4 Three Decomposition Theorems 170 3.5 Examples of Continua for Which T Is Continuous 176 3.6 T-Closed Sets 180 3.7 Applications 188 References 198 4 A Theorem of E. G. Effros 200 4.1 Topological Groups 200 4.2 Group Actions and a Theorem of Effros 204 References 215 5 Decomposition Theorems 216 5.1 Jones's Theorem 216 5.2 Detour to Covering Spaces 228 5.3 Rogers's Theorem 233 5.4 Case and Minc–Rogers Continua 243 5.5 Covering Spaces of Some Homogeneous Continua 249 References 257 6 n-Fold Hyperspaces 259 6.1 General Properties 259 6.2 Unicoherence 268 6.3 Aposyndesis 269 6.4 Arcwise Accessibility 272 6.5 Points That Arcwise Disconnect 275 6.6 C*n-Smoothness 285 6.7 Z-Sets 291 6.8 Retractions 300 6.9 Graphs 309 6.10 Cones, Suspensions and Products 314 6.11 Strong Size Maps 323 References 334 7 n-Fold Hyperspace Suspensions 338 7.1 General Properties 338 7.2 Contractibility 349 7.3 Aposyndesis 350 7.4 Local Connectedness 352 7.5 Points That Arcwise Disconnect 359 7.6 Cones, Suspensions and Products 364 7.7 Fixed Points 368 7.8 Absolute n-Fold Hyperspace Suspensions 372 7.9 Hereditarily Indecomposable Continua 376 References 379 8 Induced Maps on n-Fold Hyperspaces 381 8.1 General Maps 381 8.2 Induced Maps 398 8.3 Confluent Maps 404 8.4 Monotone Maps 409 8.5 Open Maps 417 8.6 Light Maps 427 8.7 Freely Decomposable and Strongly Freely Decomposable Maps 431 References 435 9 Questions 437 9.1 Inverse Limits 437 9.2 The Set Function T 439 9.3 Homogeneous Continua 441 9.4 n-Fold Hyperspaces 442 9.5 n-Fold Hyperspace Suspensions 443 9.6 Induced Maps on n-Fold Hyperspaces 444 References 445 Index 447 "This book is a significant companion text to the existing literature on continuum theory. It opens with background information of continuum theory, so often missing from the preceding publications, and then explores the following topics: inverse limits, the Jones set function T, homogenous continua, and n-fold hyperspaces. In this new edition of the book, the author builds on the aforementioned topics, including the unprecedented presentation of n-fold hyperspace suspensions and induced maps on n-fold hyperspaces. The first edition of the book has had a remarkable impact on the continuum theory community. After twelve years, this updated version will also prove to be an excellent resource within the field of topology".-- Prové de l'editor Front Matter ....Pages i-xvii Preliminaries (Sergio Macías)....Pages 1-51 Inverse Limits and Related Topics (Sergio Macías)....Pages 53-111 Jones’s Set Function \(\mathcal {T}\) (Sergio Macías)....Pages 113-186 A Theorem of E. G. Effros (Sergio Macías)....Pages 187-202 Decomposition Theorems (Sergio Macías)....Pages 203-245 n-Fold Hyperspaces (Sergio Macías)....Pages 247-325 n-Fold Hyperspace Suspensions (Sergio Macías)....Pages 327-369 Induced Maps on n-Fold Hyperspaces (Sergio Macías)....Pages 371-426 Questions (Sergio Macías)....Pages 427-436 Back Matter ....Pages 437-441