View source for Book:A. Illanes/Hyperspaces, Fundamentals and Recent Advances

{{Book|Hyperspaces, Fundamentals and Recent Advances|1999|Marcel Dekker||Alejandro Illanes|author2=Sam B. Nadler, Jr.}}

===Contents===
:Preface
:Part One
::I. The Topology for Hyperspaces
:::1. The General Notation of a Hyperspace
::::Topological Invariance
::::Specified Hyperspaces
::::Exercises
:::2. The Hausdorff Metric $H_d$
::::Proof that $H_d$ Is a Metric
::::A Results about Metrizability of $\mathrm{CL}(X)$
::::Exercises
:::3. Metrizability of Hyperspaces
::::Metrizability of $2^X$
::::Metrizability and Compactness of $\mathrm{CL}(X)$
::::Exercises
:::4. Convergence in Hyperspaces
::::$L$-convergence, $T_V$-convergence
::::Relationships between $L$-convergence and $T_V$-convergence
::::When $X$ Is Compact Hausdorff
::::Countable Compactness Is Necessary
::::Exercises
:::References
::II. Examples: Geometric Models for Hyperspaces
:::5. $\mathrm{C}(X)$ for Certain Finite Graphs $X$
::::$X$ an Arc
::::$X$ a Simple Closed Curve
::::$X$ a Noose
::::$X$ a Simple $n$-od
::::Historical Comments
::::Exercises
:::6. $\mathrm{C}(X)$ when $X$ Is the Hairy Point
::::Exercises
:::7. $\mathrm{C}(X)$ when $X$ is the Circle-with-a-Spiral
::::Cones, Geometric Cones
::::The Model for $\mathrm{C}(X)$
::::Knaster's Question
::::When $\mathrm{C}(Y) \approx \mathrm{Cone}(Y)$
::::Exercises
:::8. $2^X$ when $X$ Is Any Countably Infinite Compactum
::::Cantor sets
::::Preliminary Results
::::Structure Theorem
::::Uniqueness of Compactifications
::::The Model for $2^X$
::::Exercises
:::References
::III. $2^X$ and $\mathrm{C}(X)$ for Peano Continua $X$
:::9. Preliminaries: Absolute Retracts, $Z$-sets, Torunczyk's Theorem
::::Exercises
:::10. Preliminaries: General Results about Peano Continua
::::Exercises
:::11. The Curtis-Schori Theorem for $2^X$ and $\mathrm{C}(X)$
::::When $W^X_K$ and $\mathrm{C}_K(X)$ are $Z$-sets
::::The Curtis-Schori Theorem
::::Further Uses of Torunczyk's Theorem
::::Exercises
:::References
::IV. Arcs in Hyperspaces
:::12. Preliminaries: Separation, Quasicomponents, Boundary Bumping
::::Bumping
::::Exercises
:::13. A Brief Introduction to Whitney Maps
::::Definition of a Whitney Map
::::Existence of Whitney Maps
::::Exercises
:::14. Order Arcs and Arcwise Connectedness of $2^X$ and $\mathrm{C}(X)$
::::Definition of Order Arc
::::Arcwise Connectedness of $2^X$ and $\mathrm{C}(X)$
::::Application: $2^X \supset I^{\infty}$
::::Original Sources
::::Exercises
:::15. Existence of an Order Arc from $A_0$ to $A_1$
::::Necessary and Sufficient Condition
::::Application: Homogeneous Hyperspaces
::::Original Sources
::::Exercises
:::16. Kelley's Segments
::::Kelley's Notion of a Segment
::::Results about Segments
::::Addendum: Extending Whitney Maps
::::Original Sources
::::Exercises
:::17. Spaces of Segments, $S_w(\mathcal{H})$
::::Compactness
::::$S_w(\mathcal{H}) \approx \overline{\mathcal{P}}(\mathcal{H})$
::::$S_w(2^X), S_w(\mathrm{C}(X))$ When $X$ Is a Peano Continuum
::::Application: Mapping the Cantor Fan Onto $2^X$ and $\mathrm{C}(X)$
::::Original Sources
::::Exercises
:::18. When $\mathrm{C}(X)$ Is Uniquely Arcwise Connected
::::Structure of Arcs in $\mathrm{C}(X)$ When $X$ Is Hereditarily Indecomposable
::::Uniqueness of Arcs in $\mathrm{C}(X)$ When $X$ Is Hereditarily Indecomposable
::::The Characterization Theorem
::::Original Sources
::::Exercises
::::References
::V. Shape and Contractibility of Hyperspaces
:::19. $2^X$ and $\mathrm{C}(X)$ as Nested Intersections of ARs
::::$2^X, \mathrm{C}(X)$ Are Acyclic
::::$2^X, \mathrm{C}(X)$ Are crANR
::::$2^X, \mathrm{C}(X)$ Are Unicoherent
::::Whitney Levels in $\mathrm{C}(X)$ Are Continua
::::$2^X, \mathrm{C}(X)$ Have Trivial Shape
::::Original Sources
::::Exercises
:::20. Contractible Hyperspaces
::::The Fundamental Theorem
::::$X$ Contractible, $X$ Hereditarily Indecomposable
::::Property $(\kappa)$ (Kelley's Property)
::::Theorem about Property ($\kappa$)
::::$X$ Peano, $X$ Homogeneous
::::Original Sources
::::Exercises
:::References
::VI. Hyperspaces and the Fixed Point Property
:::21. Preliminaries: Brouwer's Theorem, Universal Maps, Lokuciewski's Theorem
::::Original Sources
::::Exercises
:::22. Hyperspaces with the Fixed Point Property
::::Peano Continua
::::Arc-like Continua
::::Circle-like Continua
::::A General Theorem
::::Dendroids
::::Hereditarily Indecomposable Continua
::::Addendum: $\mathrm{Dim}[\mathrm{C}(X)] \geq 2$
::::Original Sources
::::Exercises
:::References
:Part Two
::VII. Whitney Maps
:::23. Existence and Extensions
::::Exercises
:::24. Open and Monotone Whitney Maps for $2^X$
::::Exercises
:::25. Admissible Whitney Maps
::::Exercises
:::26. A Metric on Hyperspaces Defined by Whitney Maps
::::Exercises
:::References
::VIII. Whitney Propertis and Whitney-Reversible Properties
:::27. Definitions
::::Exercises
:::28. ANR
::::Exercises
:::29. Aposyndesis
::::Exercises
:::30. AR
::::Exercises
:::31. Being an Arc
::::Exercises
:::32. Arc-Smoothness
:::33. Arcwise Connectedness
::::Exercises
:::34. Being Atriodic
::::Exercises
:::35. $C^*$-Smoothness, $\mathrm{Class}(W)$ and Covering Property
::::Exercises
:::36. Cech Cohomology Groups, Acyclicity
:::37. Chainability (Arc-Likeness)
::::Exercises
:::38. Being a Circle
::::Exercises
:::39. Circle-Likeness
::::Exercises
:::40. Cone = Hyperspace Property
::::Exercises
:::41. Contractibility
::::Exercises
:::42. Convex Metric
::::Exercises
:::43. Cut Points
::::Exercises
:::44. Decomposability
::::Exercises
:::45. Dimension
::::Exercises
:::46. Fixed Point Property
::::Exercises
:::47. Fundamental Group
::::Exercises
:::48. Homogeneity
::::Exercises
:::49. Irreducibility
::::Exercises
:::50. Kelley's Property
::::Exercises
:::51. $\lambda$-Connectedness
::::Exercises
:::52. Local Connectedness
::::Exercises
:::53. $n$-Connectedness
::::Exercises
:::54. Planarity
::::Exercises
:::55. $\mathcal{P}$-Likeness
::::Exercises
:::56. Pseudo-Arc
::::Exercises
:::57. Pseudo-Solenoids and the Pseudo-Circle
::::Exercises
:::58. $R^3$-Continua
::::Exercises
:::59. Rational Continua
::::Exercises
:::60. Shape of Continua
:::61. Solenoids
:::62. Span
:::63. Tree-Likeness
:::64. Unicoherence
::::Exercises
::::Table Summarizing Chapter VIII
:::References
::IX. Whitney Levels
:::65. Finite Graphs
::::Exercises
:::66. Spaces of the Form $C_E(X,t)$ Are ARs
::::Exercises
:::67. Absolutely $C^*$-Smooth, $\mathrm{Class}(W)$ and Covering Property
::::Exercises
:::68. Holes in Whitney Levels
:::References
::X. General Properties of Hyperspaces
:::69. Semi-Boundaries
::::Exercises
:::70. Cells in Hyperspaces
::::Exercises
:::71. Neighborhoods of $X$ in the Hyperspaces
::::Exercises
:::References
::XI. Dimension of $\mathrm{C}(X)$
:::72. Previous Results about Dimension of Hyperspaces
::::Exercises
:::73. Dimension of $\mathrm{C}(X)$ for $2$-Dimensional Continua $X$
::::Exercises
:::74. Dimension of $\mathrm{C}(X)$ for $1$-Dimensional Continua $X$
:::References
::XII. Special Types of Maps between Hyperspaces
:::75. Selections
::::Exercises
:::76. Retractions between Hyperspaces
::::Exercises
:::77. Induced Maps
::::Exercises
:::References
::XIII. More on Contractibility of Hyperspaces
:::78. More on Contractible Hyperspaces
::::Contractibility vs. Smoothness in Hyperspaces
::::$R^3$-Sets
::::Spaces of Finite Subsets
::::Admissibility
::::Maps Preserving Hyperspace Contractibility
::::More on Kelley's Property
::::Exercises
:::References
::XIV. Products, Cones and Hyperspaces
:::79. Hyperspaces Which Are Products
::::Wrinkles
::::Folds
::::Proof of the Main Theorem
::::Exercises
:::80. More on Hyperspaces and Cones
::::Exercises
:::References
::XV. Questions
:::81. Unsolved and Partially Solved Questions of [56]
:::82. Solved Questions of [56]
:::83. More Questions
::::General Spaces
::::Geometric Models
::::$Z$-Sets
::::Symmetric Products
::::Size Maps
::::The Space of Whitney Levels for $2^X$
::::Aposyndesis
::::Universal Maps
:::References
:::Literature Related to Hyperspaces of Continua Since 1978
:Special Symbols
:Index

[[Category:Book]]