Continuous selections of multivalued mappings

Title  ......Page 1
Preface  ......Page 3
Contents  ......Page 6
0.1. Topological spaces. Separation axioms  ......Page 11
0.2. Topological vector spaces. Separation theorems ......Page 18
0.3. Banach spaces  ......Page 24
0.4. Extensions of continuous functions. Extensors and retracts  ......Page 30
0.5. Multivalued mappings. Types of continuity. Selections  ......Page 34
1.1. Paracompactness of domain as a necessary condition  ......Page 40
1.2. Method of outside approximations  ......Page 45
1.3. Method of inside approximations  ......Page 49
1.4. Properties of paracompact spaces. Nerves of coverings  ......Page 53
2.1. Zero-dimensionality of the domain as a necessary condition  ......Page 62
2.2. Proof of the Zero-dimensional selection theorem  ......Page 64
3.1. Preliminaries. Probabilistic measure and integration  ......Page 68
3.2. Milyutin mappings. Convex-valued selection theorem via the zero-dimensional theorem  ......Page 75
3.3. Existence of Milyutin mappings on the class of paracompact spaces  ......Page 77
3.4. Zero-dimensionality of Xo and continuity of  ......Page 79
§4 Compact-valued selection theorem  ......Page 83
4.2. Proof via inside approximations  ......Page 84
4.3. Method of coverings  ......Page 88
5.1. Cn and LCn subsets of a topological spaces. Equi-LCn and Uniformly Equi-LCn families of subsets of a topological and metric spaces  ......Page 92
5.2. Shift selection theorem. Scheme of the proof  ......Page 99
5.3. Proofs of main lemmas and Controlled Contractibility theorem  ......Page 104
5.4. Prom Nerve-Weak Shift Selection Theorem to Shift Selection Theorem  ......Page 108
5.5. Prom Controlled Extension Theorem to Nerve-Weak Shift Selection Theorem  ......Page 110
5.6. Controlled Extension Theorem  ......Page 113
5.7. Finite-Dimensional Selection Theorem. Uniform relative version  ......Page 117
5.8. Prom the UELCn restrictions to ELCn restrictions. The end of the proofs  ......Page 121
§6 Examples and counterexamples  ......Page 124
7.1. Filtered multivalued mappings. Statements of the results  ......Page 135
7.2. Singlevalued approxmations of upper semicontinuous mappings  ......Page 141
7.3. Separations of multivalued mappings. Proof of Theorem (I.7.6)  ......Page 145
7.4. Enlargements of compact-valued mappings. Proof of Theorem (I.7.7}  ......Page 149
1.1. Some other convex-valued selection theorems  ......Page 152
1.2. Characterizations via compact-valued selection theorems  ......Page 155
1.3. Dense families of selections. Characterization of perfect normality  ......Page 156
1.4. Selections of nonclosed-valued equi-LCn mappings  ......Page 159
2.1. Union of finite-dimensional and convex-valued theorems. Approximative selection properties  ......Page 162
2.2. "Countable" type selection theorems and their unions with other selection theorems  ......Page 164
3.1. Lower semicontinuous selections and derived mappings  ......Page 167
3.2. Almost lower semicontinuity  ......Page 170
3.3. Quasi lower semicontinuity  ......Page 171
3.4. Further generalizations of lower semicontinuity  ......Page 176
3.5. Examples  ......Page 178
4.1. Paraconvexity. Function of non-convexity of closed subsets of normed spaces  ......Page 181
4.2. Axiomatic definition of convex structures in metric spaces. Geodesic structures  ......Page 184
4.3. Topological convex structure  ......Page 188
4.4. Decomposable subsets of spaces of measurable functions  ......Page 192
5.1. Metrizability of the range of a multivalued mapping  ......Page 195
5.2. A weakening of the metrizability of the ranges  ......Page 200
5.3. Hyperspaces, selections and orderability  ......Page 204
5.4. Densely defined selections  ......Page 208
5.5. Continuous multivalued approximations of semicontinuous multivalued mappings  ......Page 213
5.6. Without classification  ......Page 217
5.7. Recent results  ......Page 222
6.1. Uniformization problem  ......Page 225
6.2. Measurable multivalued mappings  ......Page 229
6.3. Measurable selections of semicontinuous mappings  ......Page 233
6.4. Caratheodory conditions. Solutions of differential inclusions  ......Page 236
1.1. Extensions  ......Page 243
1.2. Bartle-Graves type theorems. Theory of liftings  ......Page 244
1.3. Homeomorphism problem for separable Banach spaces  ......Page 248
1.4. Applications of the Zero-dimensional selection theorem  ......Page 249
1.5. Continuous choice in the continuity type definitions  ......Page 251
1.6. Paracompactness of CW-complexes  ......Page 253
1.7. Miscellaneous results  ......Page 255
2.1. Dyer-Hamstrom theorem  ......Page 259
2.2. Regular mappings with fibers homeomorphic to an interval  ......Page 261
2.3. Strongly regular mappings. Ferry's theorem  ......Page 264
2.4. Non compact fibers. Exact Milyutin mappings  ......Page 266
3.1. Fixed-point theorems and fixed-points sets for convex-valued mappings  ......Page 273
3.2. Fixed-points sets for non-convex valued mappings  ......Page 277
3.3. Hubert spaces case  ......Page 279
3.4. An application of selections in the finite-dimensional case  ......Page 282
3.5. Fixed-point theorem for decomposable-valued contraction  ......Page 283
4.1. Statement of the problem. Solution of the HGP for n=1  ......Page 286
4.2. The space of all self-homeomorphisms of the disk  ......Page 289
5.1. Dugundji mappings in AE(0)-compacta  ......Page 293
5.2. Dugundji mappings and 0-soft mappings  ......Page 298
5.3. General conception of softness of mappings. Adequacy problem  ......Page 302
5.4. Parametric versions of the Wazewski-Viet oris-Wo jdislawski theorem  ......Page 304
5.5. Functor of probabilistic measures  ......Page 307
6.1. Proximinal and Cebisev subsets of normed spaces  ......Page 312
6.2. Continuity of metric projections and ?-projections  ......Page 315
6.3. Continuous selections of metric projections in spaces of continuous functions and Lp-spaces  ......Page 320
6.4. Rational e-approximations in spaces of continuous functions and Lp-spaces  ......Page 321
7.1. Decomposable sets in functional spaces  ......Page 324
7.2. Selection approach to solution of differential inclusions. First results ..  ......Page 330
7.3. Selection theorems for decomposable valued mappings  ......Page 334
7.4. Directionally continuous selections  ......Page 340
§8 References  ......Page 343
Index  ......Page 363