## Cantor Set

The Cantor set () is given by taking the interval [0,1] (set ), removing the middle third (), removing the middle third of each of the two remaining pieces (), and continuing this procedure ad infinitum. It is therefore the set of points in the Interval [0,1] whose ternary expansions do not contain 1, illustrated below.

This produces the Set of Real Numbers such that

 (1)

where may equal 0 or 2 for each . This is an infinite, Perfect Set. The total length of the Line Segments in the th iteration is
 (2)

and the number of Line Segments is , so the length of each element is
 (3)

and the Capacity Dimension is
 (4)

The Cantor set is nowhere Dense, so it has Lebesgue Measure 0.

A general Cantor set is a Closed Set consisting entirely of Boundary Points. Such sets are Uncountable and may have 0 or Positive Lebesgue Measure. The Cantor set is the only totally disconnected, perfect, Compact Metric Space up to a Homeomorphism (Willard 1970).

References

Boas, R. P. Jr. A Primer of Real Functions. Washington, DC: Amer. Math. Soc., 1996.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 15-20, 1991.