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

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## W.H. Young and Grace Chisholm Young:

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

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

### Subject Matter

### 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

- 68. Order is property of the set

- 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