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E-grāmata: Hyperspaces: Fundamentals and Recent Advances

Sam Nadler (Morgantown, West Virginia, USA), Alejandro Illanes

Formāts: 544 pages
Sērija : Chapman & Hall/CRC Pure and Applied Mathematics
Izdošanas datums: 15-Dec-2024
Izdevniecība: CRC Press Inc
Valoda: eng
ISBN-13: 9781351439428

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Formāts: 544 pages
Sērija : Chapman & Hall/CRC Pure and Applied Mathematics
Izdošanas datums: 15-Dec-2024
Izdevniecība: CRC Press Inc
Valoda: eng
ISBN-13: 9781351439428

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Provides an overview of the fundamentals of hyperspaces, then surveys a range of recent research in the field, with emphasis on the hyperspaces 2x and C(X) , where X is a continuum. Discusses symmetric products, containment hyperspaces, selections, spaces of segments, and spaces of Whitney levels, and incorporates basic material on absolute retracts, infinite-dimensional topology, Z-sets, and the fixed point property. Presents both solved and unsolved problems in hyperspaces, including some appearing in print for the first time. Some problems addressed include the dimension problem, the n -od problem, and the product problem. Annotation c. by Book News, Inc., Portland, Or.

Presents hyperspace fundamentals, offering a basic overview and a foundation for further study. Topics include the topology for hyperspaces, examples of geometric models for hyperspaces, 2x and C(X) for Peano continua X, arcs in hyperspaces, the shape and contractability of hyperspaces, hyperspaces and the fixed point property, and Whitney maps. The text contains examples and exercises throughout, and provides proofs for most results.

Preface

Part One

The Topology for Hyperspaces

(28)

The General Notion of a Hyperspace

(6)

Topological Invariance

(1)

Specified Hyperspaces

(1)

Exercises

(2)

The Hausdorff Metric Hd

(7)

Proof That Hd Is a Metric

(1)

A Result about Metrizability of CL (X)

(2)

Exercises

(2)

Metrizability of Hyperspaces

(4)

Metrizability of 2x

(2)

Metrizability and Compactness of CL(X)

(1)

Exercises

(1)

Convergence in Hyperspaces

(11)

L-convergence, Tv-convergence

(2)

Relationships between L-convergence and Tv-convergenece

(3)

When X Is Compact Hausdorff

(1)

Countable Compactness Is Necessary

(1)

Exercises

(2)

References

(3)

Examples: Geometric Models for Hyperspaces

(44)

C(X) for Certain Finite Graphs X

(13)

X an Arc

(2)

X a Simple Closed Curve

(1)

X a Noose

(3)

X a Simple n-od

(5)

Historical Comments

(1)

Exercises

(2)

C(X) When X Is the Hairy Point

(4)

Exercises

(14)

C(X) When X Is the Circle-with-a-Spiral

(1)

Cones, Geometric Cones

(2)

The Model for C(X)

(6)

Knaster's Question

(1)

When C(Y) ≈ Cone(Y)

(3)

Exercises

(2)

2X When X Is Any Countably Infinite Compactum

(11)

Cantor Sets

(1)

Preliminary Results

(2)

Structure Theorem

(1)

Uniqueness of Compactifications

(3)

The Model for 2X

(2)

Exercises

(1)

References

(3)

2X and C(X) for Peano Continua X

(22)

Preliminaries: Absolute Retracts, Z-sets, Torunczyk's Theorem

(4)

Exercises

(1)

Preliminaries: General Results about Peano Continua

(5)

Exercises

(2)

The Curtis-Schori Theorem for 2X and C(X)

(12)

When 2XK and CK(X) Are Z-sets

(4)

The Curtis-Schori Theorem

(1)

Further Uses of Torunczyk's Theorem

(1)

Exercises

(3)

References

(3)

Arcs in Hyperspaces

(56)

Preliminaries: Separation, Quasicomponents, Boundary Bumping

(8)

Exercises

103

(2)

A Brief Introduction to Whitney Maps

105

(5)

Definition of a Whitney Map

105

(1)

Existence of Whitney Maps

106

(2)

Exercises

108

(2)

Order Arcs and Arcwise Connectedness of 2X and C(X)

110

(9)

Definition of Order Arc

110

(1)

Arcwise Connectedness of 2X and C(X)

110

(4)

Application: 2X ⊃ I∞

114

(2)

Original Sources

116

(1)

Exercises

117

(2)

Existence of an Order Arc from A0 to A1

119

(8)

Necessary and Sufficient Condition

119

(3)

Application: Homogeneous Hyperspaces

122

(2)

Original Sources

124

(1)

Exercises

124

(3)

Kelley's Segments

127

(7)

Kelley's Notion of a Segment

127

(1)

Results about Segments

128

(3)

Addendum: Extending Whitney Maps

131

(1)

Original Sources

132

(1)

Exercises

132

(2)

Spaces of Segments, Sw(H)

134

(9)

Compactness

134

(2)

SW(H) ≈ O(H)

136

(2)

Sw(2X), SW(C(X)) When X Is a Peano Continuum

138

(2)

Application: Mapping the Cantor Fan Onto 2X and C(X)

140

(1)

Original Sources

141

(1)

Exercises

141

(2)

When C(X) Is Uniquely Arcwise Connected

143

(10)

Structure of Arcs in C(X) When X Is Hereditarily Indecomposable

145

(1)

Uniqueness of Arcs in C(X) When X Is Hereditarily Indecomposable

146

(2)

The Characterization Theorem

148

(1)

Original Sources

148

(1)

Exercises

148

(1)

References

149

(4)

Shape and Contractibility of Hyperspaces

153

(28)

2X and C(X) as Nested Intersections of ARs

153

(11)

2X, C(X) Are Acyclic

155

(1)

2X, C(X) Are crANR

155

(2)

2X, C(X) Are Unicoherent

157

(2)

Whitney Levels in C(X) Are Continua

159

(1)

2X, C(X) Have Trivial Shape

160

(1)

Original Sources

161

(1)

Exercises

161

(3)

Contractible Hyperspaces

164

(17)

The Fundamental Theorem

164

(2)

X Contractible, X Hereditarily Indecomposable

166

(1)

Property (k) (Kelley's Property)

167

(1)

Theorem about Property (k)

168

(5)

X Peano, X Homogeneous

173

(2)

Original Sources

175

(1)

Exercises

176

(1)

References

177

(4)

Hyperspaces and the Fixed Point Property

181

(24)

Preliminaries: Brouwer's Theorem, Universal Maps, Lokuciewski's Theorem

181

(6)

Original Sources

186

(1)

Exercises

186

(1)

Hyperspaces with the Fixed Point Property

187

(18)

Peano Continua

187

(1)

Arc-like Continua

187

(3)

Circle-like Continua

190

(2)

A General Theorem

192

(1)

Dendroids

193

(3)

Hereditarily Indecomposable Continua

196

(1)

Addendum: Dim[ C(X)] ≥ 2

196

(1)

Original Sources

197

(1)

Exercises

197

(2)

References

199

(6)

Part Two

Whitney Maps

205

(26)

Existence and Extensions

205

(2)

Exercises

206

(1)

Open and Monotone Whitney Maps for 2X

207

(9)

Exercises

215

(1)

Admissible Whitney Maps

216

(11)

Exercises

225

(2)

A Metric on Hyperspaces Defined by Whitney Maps

227

(4)

Exercises

227

(1)

References

228

(3)

Whitney Properties and Whitney-Reversible Properties

231

(74)

Definitions

231

(3)

Exercises

233

(1)

ANR

234

(4)

Exercises

236

(2)

Aposyndesis

238

(1)

Exercises

239

(1)

239

(6)

Exercises

245

(1)

Being an Arc

245

(2)

Exercises

246

(1)

Arc-Smoothness

247

(1)

Arcwise Connectedness

247

(4)

Exercises

251

(1)

Being Atriodic

251

(2)

Exercises

253

(1)

C*-Smoothness, Class(W) and Covering Property

253

(4)

Exercises

256

(1)

Cech Cohomology Groups, Acyclicity

257

(1)

Chainability (Arc-Likeness)

257

(2)

Exercises

259

(1)

Being a Circle

259

(1)

Exercises

259

(1)

Circle-Likeness

259

(2)

Exercises

260

(1)

Cone = Hyperspace Property

261

(1)

Exercise

262

(1)

Contractibility

262

(3)

Exercises

264

(1)

Convex Metric

265

(1)

Exercises

265

(1)

Cut Points

265

(2)

Exercises

267

(1)

Decomposability

267

(1)

Exercises

268

(1)

Dimension

268

(2)

Exercises

269

(1)

Fixed Point Property

270

(1)

Exercises

270

(1)

Fundamental Group

271

(1)

Exercises

271

(1)

Homogeneity

271

(2)

Exercises

272

(1)

Irreducibility

273

(3)

Exercises

276

(1)

Kelley's Property

276

(3)

Exercises

278

(1)

λ-Connectedness

279

(2)

Exercises

280

(1)

Local Connectedness

281

(1)

Exercises

281

(1)

n-Connectedness

281

(2)

Exercises

283

(1)

Planarity

283

(1)

Exercises

284

(1)

P-Likeness

284

(2)

Exercises

285

(1)

Pseudo-Arc

286

(1)

Exercise

286

(1)

Pseudo-Solenoids and the Pseudo-Circle

286

(1)

R3-Continua

286

(1)

Exercise

287

(1)

Rational Continua

287

(1)

Exercises

287

(1)

Shape of Continua

287

(3)

Solenoids

290

(1)

Span

291

(1)

Tree-Likeness

292

(1)

Unicoherence

292

(13)

Exercises

293

(1)

Table Summarizing
Chapter VIII

294

(5)

References

299

(6)

Whitney Levels

305

(28)

Finite Graphs

305

(9)

Exercises

313

(1)

Spaces of the Form CE (X,t) Are ARs

314

(5)

Exercises

318

(1)

Absolutely C*-Smooth, Class(W) and Covering Property

319

(7)

Exercises

325

(1)

Holes in Whitney Levels

326

(7)

References

329

(4)

General Properties of Hyperspaces

333

(14)

Semi-Boundaries

333

(4)

Exercises

336

(1)

Cells in Hyperspaces

337

(5)

Exercises

341

(1)

Neighborhoods of X in the Hyperspaces

342

(5)

Exercises

344

(1)

References

345

(2)

Dimension of C(X)

347

(16)

Previous Results about Dimension of Hyperspaces

347

(2)

Exercises

348

(1)

Dimension of C(X) for 2-Dimensional Continua X

349

(9)

Exercises

357

(1)

Dimension of C(X) for 1-Dimensional Continua X

358

(5)

References

359

(4)

Special Types of Maps between Hyperspaces

363

(32)

Selections

363

(8)

Exercises

368

(3)

Retractions between Hyperspaces

371

(10)

Exercises

379

(2)

Induced Maps

381

(14)

Exercises

387

(3)

References

390

(5)

E-grāmata: Hyperspaces: Fundamentals and Recent Advances

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