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

Subject Matter

• Set Theory

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

28. Adherences and coherences

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

53. The (inner) addition Theorem

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

65. The (outer) additive class

66. The additive class

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