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

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