Editing Smith–Volterra–Cantor set (section)

==Construction==

Similar to the construction of the [[Cantor set]], the Smith–Volterra–Cantor set is constructed by removing certain intervals from the [[unit interval]] <math>[0, 1].</math> 

The process begins by removing the middle 1/4 from the interval <math>[0, 1]</math> (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is 
<math display=block>\left[0, \tfrac{3}{8}\right] \cup \left[\tfrac{5}{8}, 1\right].</math>

The following steps consist of removing subintervals of width <math>1/4^n</math> from the middle of each of the <math>2^{n-1}</math> remaining intervals.  So for the second step the intervals <math>(5/32, 7/32)</math> and <math>(25/32, 27/32)</math> are removed, leaving 
<math display=block>\left[0, \tfrac{5}{32}\right] \cup \left[\tfrac{7}{32}, \tfrac{3}{8}\right] \cup \left[\tfrac{5}{8}, \tfrac{25}{32}\right] \cup \left[\tfrac{27}{32}, 1\right].</math>

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. 

[[Image:Smith-Volterra-Cantor set.svg|center|512px]]

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 Smith–Volterra–Cantor set has positive measure while the Cantor set has zero measure.