The Cantor set is a subset of the real line built by starting with the interval from 0 to 1 and repeatedly removing the open middle third of every remaining segment, infinitely many times. Georg Cantor described the construction in 1883 as an example of a set that is nowhere dense yet contains no isolated points. Despite being built by removing almost everything, the Cantor set is uncountably infinite, in one-to-one correspondence with the real numbers themselves, while its total length, or Lebesgue measure, is exactly zero. It is self-similar, consisting of two scaled copies of itself joined end to end, making it a foundational example of a fractal long before that word existed.
Facts
FAQs
Which numbers belong to the Cantor set?
Exactly the numbers from 0 to 1 that can be written in base three without using the digit 1.
The Cantor set is the set of numbers in the interval from 0 to 1 whose ternary expansions, that is their base three expansions, do not contain the digit 1. Equivalently, every member can be written using only the digits 0 and 2. It is also called the Cantor comb or the no middle third set, because it is built by repeatedly removing the open middle third of each remaining piece of the interval.
Connections
Associated With
Georg Cantor introduced the set in 1883 as an example within his work on point-set topology and the theory of trigonometric series.
Source Cantor Set (Wikipedia)
Attributed To
Source Cantor Set (Wikipedia)
In Branch
Source Cantor Set (Wikipedia)
Is Kind Of Object
Sources
1. Cantor Set (Wikipedia)
Wikimedia FoundationLead section
In mathematics, the Cantor set is a self-similar set of points lying on a single line segment that has a number of unintuitive properties.
- In Branch: Set Theory
- Associated With: Georg Cantor
Attributed To: Henry John Stephen Smith, Lead
It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883.
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