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

Subject Matter

 * Set Theory

Contents

 * Preface (,, 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