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) |

(2) |

(3) |

(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.

Willard, S. §30.4 in *General Topology.* Reading, MA: Addison-Wesley, 1970.

© 1996-9

1999-05-26