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Concept
Cantor set
Summary
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.
More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). By a theorem of L. E. J. Brouwer, this is equivalent to being perfect nonempty, compact metrizable and zero dimensional.
The Cantor ternary set is created by iteratively deleting the open middle third from a set of line segments. One starts by deleting the open middle third from the interval , leaving two line segments: . Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: .
The Cantor ternary set contains all points in the interval that are not deleted at any step in this infinite process. The same facts can be described recursively by setting
and
for , so that
for any .
The first six steps of this process are illustrated below.
Using the idea of self-similar transformations, and the explicit closed formulas for the Cantor set are
where every middle third is removed as the open interval from the closed interval surrounding it, or
where the middle third of the foregoing closed interval is removed by intersecting with
This process of removing middle thirds is a simple example of a finite subdivision rule. The complement of the Cantor ternary set is an example of a fractal string.
Official source
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