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

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Alejandro Illanes and Sam B. Nadler, Jr.: Hyperspaces, Fundamentals and Recent Advances

Published $1999$, Marcel Dekker.


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