Cantor set

Lore & Background

The most common construction is the Cantor ternary set, built by deleting the open middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense. The Cantor ternary set is created by iteratively deleting the open middle third from a set of line segments, starting with the interval [0,1] and removing (1/3, 2/3), leaving two segments, then continuing indefinitely.

Reader's Guide

The Cantor set is significant because it helped lay the foundations of modern point-set topology. In topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product of the discrete two-point space. By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact, metrizable and zero-dimensional. The set's self-similar nature and its contrast with a linear continuum make it a fundamental example in mathematics, illustrating concepts such as nowhere denseness and perfect sets.

Did You Know?

• The Cantor set was discovered in 1874 by Henry John Stephen Smith, not by Cantor.
• Cantor mentioned the ternary construction only in passing, as an example of a perfect set that is nowhere dense.
• The Cantor set is naturally homeomorphic to the countable product of the discrete two-point space.
• By a theorem of L. E. J. Brouwer, the Cantor set is equivalent to being perfect, nonempty, compact, metrizable and zero-dimensional.