Cantor Set, Mu-Ency at MROB

Cantor Set

Robert P. Munafo, 2008 Feb 1.

math. 1. A set of points, no two of which are touching, but each of which is a limit point of other points in the set.

2. the Cantor middle-thirds set: A set of points on a line segment that is the result of an infinite number of steps, in which each step consists of removing the middle-third of the segment(s) remaining from the previous step. It has a Hausdorff dimension of 0.631:

step 0: _________________________________________________________________________________
step 1: ___________________________ ___________________________
step 2: _________ _________ _________ _________
step 3: ___ ___ ___ ___ ___ ___ ___ ___
step 4: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(etc.)

A Cantor set has an infinite ("uncountable") number of points, the same number as a line segment or any other continuum. However (and paradoxically, by the intuition of Cantor's time) it has zero Lebesgue measure.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index