The Theory of Sets of Points

Cover......Page 1
The Theory of Sets of Points......Page 4
PREFACE......Page 10
Contents......Page 12
ART. 2. Sets and sequences......Page 18
ART. 4. Magnitude and equality......Page 20
ART. 5. The number [INFINITY]......Page 22
ART. 7. Algebraic and transcendental numbers......Page 23
ART. 8. The projective scale......Page 26
ART. 9. Interval between two numbers......Page 31
ART. 10. Sets of points. Sequences. Limiting points......Page 33
ART. 11. Fundamental sets......Page 39
ART. 12. Derived sets. Limiting points of various orders......Page 40
ART. 13. Deduction......Page 42
ART. 14. Theorems about a set and its derived and deduced sets......Page 44
ART. 15. Intervals and their limits......Page 46
ART. 16. Upper and lower limit......Page 48
ART. 17. Measurement and potencies......Page 50
ART. 18. Countable sets......Page 51
ART. 19. Preliminary definitions of addition and multiplication......Page 53
ART. 20. Coiintable sets of intervals......Page 55
ART. 22. More than countable sets......Page 59
ART. 23. The potency c......Page 63
ART. 24. Symbolic equations involving the potency c......Page 67
ART. 25. Limiting points of countable and more than countable degree......Page 70
ART. 27. Derived and deduced sets......Page 72
ART. 28. Adherences and coherences......Page 74
ART. 29. The ultimate coherence......Page 78
ART. 30. Tree illustrating the theory of adherences and coherences......Page 79
ART. 31. Ordinary inner limiting sets......Page 80
ART. 32. Relation of any set to the inner limiting set of a series of sets of intervals containing the given set......Page 82
ART. 33. Generalised inner and outer limiting sets......Page 86
ART. 34. Sets of the first and second category......Page 87
ART. 35. Generality of the class of inner and outer limiting sets......Page 89
ART. 36. Meaning of content......Page 93
ART. 39. Definition of content of such a set of intervals......Page 94
ART. 40. Examples of such sets of intervals......Page 96
ART. 41. Content of such a set and potency of complementary set of points 80......Page 97
ART. 43. Addition Theorem for the content of sets of intervals......Page 98
ART. 44. Content of a closed set of points......Page 99
ART. 45. Addition Theorem for the content of closed sets of points......Page 100
ART. 47. Historical note on the theory of content......Page 102
ART. 48, 49. Content of any closed component of an ordinary inner limiting set......Page 106
ART. 50. Content of any closed component of a generalised inner limiting set, defined by means of closed sets......Page 110
ART. 51. Open sets......Page 111
ART. 52. The (inner) content......Page 113
ART. 53. The (inner) addition Theorem......Page 114
ART. 55. The (inner) additive class, and the addition theorem for the (inner) contents......Page 115
ART. 56. Reduction of the classification of open sets to that of sets of zero (inner) content......Page 119
ART. 57. The (outer) content......Page 120
ART. 58. Measurable sets......Page 122
ART. 59. An ordinary inner or outer limiting set is measurable......Page 124
ART. 60. The (inner) additive class consists of measurable sets......Page 126
ART. 62. Outer and inner limiting sets of measurable sets......Page 127
ART. 63. Theorem for the (outer) content analogous to Theorem 20 of § 52......Page 131
ART. 65. The (outer) additive class......Page 133
ART. 66. The additive class......Page 135
ART. 67. Content of the irrational numbers......Page 136
ART. 69. The characteristic of order......Page 138
ART. 71. Order of the natural numbers......Page 139
ART. 72. Orders of closed sequences, etc......Page 140
ART. 73. Graphical and numerical representation......Page 141
ART. 74. The rational numbers. Close order......Page 144
ART. 75. Condition that a set in close order should be dense everywhere......Page 147
ART. 76. Limiting points of a set in close order......Page 149
ART. 77. Ordinally closed, dense in itself, perfect. Ordinal limiting point.......Page 150
ART. 78. Order of the continuum......Page 151
ART. 79. Order of the derived and deduced sets......Page 152
ART. 80. Well-ordered sets......Page 154
ART. 81. Multiple order......Page 160
ART. 83. General definition of the word \"set\"......Page 162
ART. 84. The Cantor-Bernstein-Schroeder Theorem......Page 164
ART. 85. Greater, equal and less......Page 166
ART. 86. The addition and multiplication of potencies......Page 167
ART. 88. Transfinite ordinals of the second class......Page 170
ART. 89. Ordinals of higher classes......Page 173
ART. 91. The theory of ordinal addition......Page 174
ART. 92. The law of ordinal multiplication......Page 176
ART. 94. The two-fold continuum......Page 178
ART. 96. Cantor\'s (1, 1)-correspondence between the points of the plane, or n-dimensional space and those of the straight line......Page 179
ART. 97. Analogous correspondence when the space is of a countably infinite number of dimensions......Page 180
ART. 98. Continuous representation......Page 181
ART. 99. Peano\'s continuous representation of the points of the unit square on those of the unit segment......Page 182
ART. 100. Discussion of the term \"space-filling curve\"......Page 184
ART. 101. Moore\'s crinkly curves......Page 185
ART. 102. Continuous (1, 1)-correspondence between the points of the whole plane and those of the interior of a circle of radius unity......Page 186
ART. 104. Limiting points, isolated points, sequences, etc. Examples of plane perfect sets......Page 187
ART. 106. The minimum distance between two sets of points......Page 192
ART. 107. Plane elements......Page 194
ART. 108. Primitive triangles......Page 196
ART. 109. Definitions of a domain, a region, etc......Page 197
ART. 111. Ordinary external points and external boundary points......Page 198
ART. 112. Describing a region......Page 199
ART. 114. The Chow......Page 200
ART. 115. The rim......Page 202
ART. 116. Sections of a region......Page 203
ART. 117. The span......Page 204
ART. 119. Case when the inner limiting set of a series of regions is a point, or a stretch......Page 205
ART. 120. Weierstrass\'s Theorem......Page 208
ART. 121. 122. General discussion of the inner limiting set of a series of regions......Page 209
ART. 123. Finite and infinite regions......Page 212
ART. 125. The rim is a perfect set dense nowhere......Page 213
ART. 126. Sets of regions......Page 214
ART. 127. Classification of the points of the plane with reference to a set of regions......Page 215
ART. 128. Cantor\'s Theorem of non-overlapping regions. The extended Heine-Borel Theorem, etc......Page 216
ART. 129. The black regions of a closed set......Page 220
ART. 130. Connected sets......Page 221
ART. 132. Simple polygonal regions......Page 223
ART. 133. The outer rim......Page 226
ART. 134. General form of a region......Page 229
ART. 135. The black region of a closed set containing no curves......Page 232
ART. 136. A continuous (1, 1)-correspondence between the points of a region of the plane and a segment of the straight line is impossible......Page 233
ART. 137. Uniform continuity......Page 235
ART. 138. Definition and fundamental properties of a curve......Page 236
ART. 139. Branches, end-points and closed curves......Page 237
ART. 140. Jordan curves......Page 239
ART. 141. Sets of arcs and closed sets of points on a Jordan curve......Page 246
ART. 143. Countable sets......Page 250
ART. 144. The potency c......Page 251
ART. 146. Ordinary inner limiting sets......Page 252
ART. 147. Relation of any set to the inner limiting set of a series of sets of regions containing the given set......Page 254
ART. 149. The theory of plane content in the piano......Page 255
ART. 150. Content of triangles and regions......Page 256
ART. 151. Content of a closed set......Page 258
ART. 152. Area of a region......Page 259
ART. 153. A simply connected non-quadrable region, whose rim is a Jordan curve of positive content......Page 261
ART. 154. Connexion between the potency of a closed set and the content of its black regions......Page 264
ART. 156. Content of any closed component of an ordinary inner limiting set......Page 266
ART. 158. Calculation of the plane content of closed sets......Page 267
ART. 159. Upper and lower m-ple and M-fold integrals......Page 268
ART. 160. Upper and lower semi-continuous functions......Page 269
ART. 161. The associated limiting functions of a function......Page 270
ART. 162. Calculation of the upper integral of an upper semi-continuous function......Page 271
ART. 163. Application of §§ 159--162 to the calculation of the content by integration......Page 273
ART. 164. Condition that a plane closed set should have zero content......Page 276
ART. 165. Expression for the content of a closed set as a generalised or Lebesgue integral......Page 277
ART. 166. Calculation of the content of any measurable set......Page 279
ART. 167. Length of a Jordan curve......Page 281
ART. 168. Calculation of the length of a Jordan curve......Page 285
ART. 169. Linear content on a rectifiable Jordan curve......Page 286
ART. 170. General notions on the subject of linear content......Page 287
ART. 171. Definition of linear content......Page 288
ART. 173. Linear content of a finite arc of a rectifiable Jordan curve......Page 290
ART. 174. Linear content of a set of arcs on a rectifiable Jordan curve......Page 291
ART. 175. Linear content of a countable closed set of points......Page 293
APPENDIX......Page 301
BIBLIOGRAPHY......Page 312
INDEX OF PROPER NAMES......Page 327
GENERAL INDEX......Page 328