# Book:W.H. Young/The Theory of Sets of Points

## W.H. Young and Grace Chisholm Young: The Theory of Sets of Points

Published $\text {1906}$, Cambridge: at the University Press.

### Contents

Preface (W.H. Young, Heswall, May 1906.)
Chapter $\text {I}$. Rational and Irrational Numbers
1. Introductory
2. Sets and sequences
3. Irrational numbers
4. Magnitude and equality
5. The number $\infty$
6. Limit
7. Algebraic and transcendental numbers

Chapter $\text {II}$. Representation of Numbers on the Straight Line
8. The projective scale
9. Interval between two numbers

Chapter $\text {III}$. The Descriptive Theory of Linear Sets of Points
10. Sets of points. Sequences. Limiting points
11. Fundamental sets
12. Derived sets. Limiting points of various orders
13. Deduction
14. Theorems about a set and its derived and deduced sets
15. Intervals and their limits
16. Upper and lover limit

Chapter $\text {IV}$. Potency, and the Generalised Idea of a Cardinal Number
17. Measurement and potencies
18. Countable sets
19. Preliminary definitions of addition and multiplication
20. Countable sets of intervals
21. Some theorems about countable sets of points
22. More than countable sets
23. The potency $c$
24. Symbolic equations involving the potency $c$
25. Limiting points of countable and more than countable degree
26. Closed and perfect sets
27. Derived and deduced sets
29. The ultimate coherence
30. Tree illustrating the theory of adherences and coherences
31. Ordinary inner limiting sets
32. Relation of any set to the inner limiting set of a series of sets of intervals containing the given set
33. Generalised inner and outer limiting sets
34. Sets of the first and second category
25. Generality of the class of inner and outer limiting sets

Chapter $\text {V}$. Content
36. Meaning of content
37. Content of a finite number of non-overlapping intervals
38. Extension to an infinite number of non-overlapping intervals
39. Definition of content of such a set of intervals
40. Examples of such sets of intervals
41. Content of such a set and potency of complementary set of points
42. Properties of the content of such a set of intervals
43. Addition Theorem for the content of sets of intervals
44. Content of a closed set of points
45. Addition Theorem for the content of a closed set of points
46. Connexion between the content and the potency
47. Historical note on the theory of content
48, 49. Content of any closed component of an ordinary inner limiting set
50. Content of any closed component of a generalised inner limiting set defined by means of closed sets
51. Open sets
52. The (inner) content
54. Possible extension of the (inner) addition Theorem
55. The (inner) additive class, and the addition Theorem for the (inner) contents
56. Reduction of the classification of open sets to that of sets of zero (inner) content
57. The (outer) content
58. Measurable sets
59. An ordinary inner or outer limiting set is measurable
60. The (inner) additive class consists of measurable sets
61. The (outer) additive class consists of measurable sets
62. Outer and inner limiting sets of measurable sets
63. Theorem for the (outer) content analogous to Theorem $20$ of $\S 52$
64. Connexion of the (outer) content with the theory of adherences and coherences
67. Content of the irrational numbers

Chapter $\text {VI}$. Order
68. Order is property of the set per se
69. The characteristic of order
70. Finite ordinal types
71. Order of the natural numbers
72. Orders of closed sequences, etc.
73. Graphical and numerical representation
74. The rational numbers. Close order
75. Condition that a set in close order should be dense everywhere
76. Limiting points of a set in close order
77. Ordinally closed, dense in itself, perfect. Ordinal limiting point
78. Order of the continuum
79. Order of the derived and deduced sets
80. Well-ordered sets
81. Multiple order

Chapter $\text {VII}$. Cantor's Numbers
82. Cardinal numbers
83. General definition of the word "set"
84. The Cantor-Bernstein-Schroeder Theorem
85. Greater, equal and less
86. The addition and multiplication of potencies
87. The Alephs
88. Transfinite ordinals of the second class
89. Ordinals of higher classes
90. The series of Alephs
91. The theory of ordinal addition
92. The law of ordinal multiplication

Chapter $\text {VIII}$. Preliminary Notions of Plane Sets
93. Space of any countable number of dimensions as fundamental region
94. The two-fold continuum
95. Dimensions of the fundamental region
96. Cantor's $\tuple {1, 1}$-correspondence between the points of the plane, or $n$-dimensional space and those of the straight line
97. Analogous correspondence when the space is of a countably infinite number of dimensions
98. Continuous representation
99. Peano's continuous representation of the points of the unit square on those of a unit segment
100. Discussion of the term "space-filling curve"
101. Moore's crinkly curves
102. Continuous $\tuple {1, 1}$-correspondence between the points of the whole plane and those of the interior of a circle of radius unity
103. Definition of a plane set of points
104. Limiting points, isolated points, sequences etc. Examples of plane perfect sets
105. Plane sequences in any set corresponding to any limiting point
106. The minimum distance between two sets of points

Chapter $\text {IX}$. Regions and Sets of Regions
107. Plane elements
108. Primitive triangles
109. Definitions of a domain, a region, etc.
110. Internal and external points of a region. Boundary and edge points.
111. Ordinary external points and external boundary points
112. Describing a region
113. Two internal points of a region can be joined by a finite set of generating triangles
114. The Chow
115. The rim
116. Sections of a region
117. The span
118. Discs
119. Case when the inner limiting set of a series of regions is a point or a stretch
120. Weierstrass's Theorem
121, 122. General discussion of the inner limiting set of a series of regions
123. Finite and infinite regions
124. The domain as space element
125. The rim is a perfect set dense nowhere
126. Sets of regions
127. Classification of the points of the plane with references to a set of regions
128. Cantor's Theorem of non-overlapping regions. The extended Heine-Borel Theorem, etc.
129. The black regions of a closed set
130. Connected sets
131. The inner limiting set of a series of regions, if dense nowhere, is a curve
132. Simple poygonal regions
133. The outer rim
134. General form of a region
135. The black region of a closed set containing no curves
136. A continuous $\tuple {1, 1}$-correspondence between the points of a region of the plane and a segment of the straight line is impossible
137. Uniform conformity

Chapter $\text {X}$. Curves
138. Definition and fundamental properties of a curve
139. Branches, end-points and closed curves
140. Jordan curves
141. Sets of arcs and closed sets of points on a Jordan curve

Chapter $\text {XI}$. Potency of Plane Sets
142. The only potencies in space of a countable number of dimensions are those which occur on a straight line
143. Countable sets
144. The potency $c$
145. Limiting points of countable and more than countable degree
146. Ordinary inner limiting sets
147. Relation of any set to the inner limiting set of a series of sets of regions containing the given set

Chapter $\text {XII}$. Plane Content and Area
148. Various kinds of content which occur in space of more than one dimension
149. The theory of plane content in the plane
150. Content of triangles and regions
151. Content of a closed set
152. Area of a region
153. A simply connected non-quadrable region, whose rim is a Jordan curve of positive content
154. Connexion between the potency of a closed set and the content of its black regions
155. Content of a countable closed set is zero
156. Content of any closed component of an ordinary limiting set
157. Measurable sets. Inner and outer measures of the content
158. Calculation of the plane content of closed sets
159. Upper and lower $n$-ple and $n$-fold integrals
160. Upper and lower semi-continuous functions
161. The associated limiting functions of a function
162. Calculation of the upper integral of an upper semi-continuous function
163. Application of $\S \S \ 159 - 162$ to the calculation of the content by integration
164. Condition that a plane closed set should have zero content
165. Expression for the content of a closed set as a generalised or Lebesgue integral
166. Calculation of the content of any measurable set

Chapter $\text {XIII}$. Length and Linear Content
167. Length of a Jordan curve
168. Calculation of the length of a Jordan curve
169. Linear content on a rectifiable Jordan curve
170. General notions on the subject of linear content
171. Definition of linear content
172. Alternative definition of linear content
173. Linear content of a finite arc of a rectifiable Jordan curve
174. Linear content of a set of arcs on a rectifiable Jordan curve
175. Linear content of a countable closed set of points

Appendix
Bibliography
Index of Proper Names
General Index