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In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set〔Aliprantis and Burkinshaw (1981), Principles of Real Analysis〕 is an example of a set of points on the real line ℝ that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. The Smith-Volterra-Cantor set is topologically equivalent to the middle-thirds Cantor set. == Construction == Similar to the construction of the Cantor set, the Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval (). The process begins by removing the middle 1/4 from the interval () (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is : The following steps consist of removing subintervals of width 1/22''n'' from the middle of each of the 2''n''−1 remaining intervals. So for the second step the intervals (5/32, 7/32) and (25/32, 27/32) are removed, leaving : Continuing indefinitely with this removal, the Smith–Volterra–Cantor set is then the set of points that are never removed. The image below shows the initial set and five iterations of this process. Each subsequent iterate in the Smith–Volterra–Cantor set's construction removes proportionally less from the remaining intervals. This stands in contrast to the Cantor set, where the proportion removed from each interval remains constant. Thus, the former has positive measure, while the latter zero measure. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Smith–Volterra–Cantor set」の詳細全文を読む スポンサード リンク
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