In mathematics, the Smith-Volterra-Cantor set is a set of points on the real line R satisfying the following interesting combination of properties:

• SVC is nowhere dense (in particular it contains no intervals),
• SVC has positive measure.

It is a close relative of the Cantor set, constructed by removing certain intervals from the unit interval [0, 1].

The process begins by removing the middle 1/4 from the interval [0, 1] (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is

.

The second step consists of removing the middle quarter of these two sets (the middle "quarters" now of size 1/64), and so on. More technically, this is done at each step by considering dyadic fractions of the form

contained in [0, 1] where a is an odd number. These fractions are then taken to be midpoints of the remaining subintervals, and

is then removed from either side of each of these midpoints (removing a total of around each such point).

It can also easily be shown to have a positive measure of 1/2 by observing that intervals of total length

are removed from [0, 1] during the construction.

The SVC is used in the construction of Volterra's function (see external link).

External link

• http://www.macalester.edu/~bressoud/talks/Volterra-4.pdf