ContentsPreface by J. van Mill iii Preface v Overview 1 1 Basic Theory 71.1 Preliminaries 71.2 Several theorems in general topology 141.3 ARs and ANRs 241.4 Z-sets and strong Z-sets 341.5 Homotopy denseness and SDAP 381.6 Absorbing sets and coabsorbing sets 42Contents Preface by J. van Mill iii Preface v Overview 1 1 Basic Theory 7 1.1 Preliminaries 7 1.2 Several theorems in general topology 14 1.3 ARs and ANRs 24 1.4 Z-sets and strong Z-sets 34 1.5 Homotopy denseness and SDAP 38 1.6 Absorbing sets and coabsorbing sets 42 1.7 Characterizations for some classical spaces 53 Note for Chapter 1 58 2 Topological Structures of Hyperspaces 61 2.1 Hyperspaces 61 2.2 Hyperspace theorem for Vietoris topology 67 2.3 Hyperspace theorem for Fell topology 71 2.4 Supplements and problems about hyperspaces 76 Note for Chapter 2 79 3 Function Spaces with Endograph Fell Topology 81 3.1 Properties of function spaces with Endograph Fell topology 81 3.2 Conditions for being metrizable of USC(X,I) 93 3.3 Conditions for being metrizable of continuous function spaces 97 3.4 A compacti?cation of metrizable continuous function space 106 3.5 Borel complexity and Baire property of space of continuous functions 111 3.6 Strong universality of continuous function space, I 121 3.7 Strong universality of continuous function space, II 132 3.8 Topological structures of function spaces 141 3.9 Remarks and problems about continuous maps 142 3.10 Appendix: Baire property 145 Note for Chapter 3 151 4 Topological Structures of Spaces of Fuzzy Numbers 153 4.1 Metrics on set of fuzzy numbers 154 4.2 Compactness and completeness of spaces of fuzzy numbers 160 4.3 Spaces of fuzzy numbers are ARs 167 4.4 Topological structures with endograph metric and Lp metrics 176 4.5 Topological structures with sendograph metric and L∞ metrics, I 186 4.6 Topological structures with sendograph metric and L∞ metrics, II 192 4.7 Remarks and problems about fuzzy numbers 201 Note for Chapter 4 204 5 Function Spaces Coming from Probability Theory 207 5.1 Spaces of copulas and subcopulas 207 5.2 Spaces of copulas and exchange copulas 216 5.3 Topological structure of space of subcouplas 221 5.4 Remarks and problems about spaces of copulas 226 Note for Chapter 5 230 6 Function Spaces Coming from Dynamical Systems 233 6.1 Box maps on closed intervals 233 6.2 Function spaces related to topological entropy 239 6.3 Topological structures of function spaces of transitive maps 251 6.4 Remarks and problems about spaces related to dynamical systems 261 Note for Chapter 6 263 7 Function Spaces Coming from Metric Measure Spaces 265 7.1 Basic knowledge on metric measure spaces 265 7.2 Topological structures of spaces of uniformly continuous functions 267 7.3 Pair of spaces on metric measure spaces 275 7.4 Remarks and problems about metric measure spaces 281 Note for Chapter 7 281 References 283 Index on Symbols 297 Index on Subjects 301