Definition:Cantor Set
Definition
As a Limit of Intersections
Define, for $n \in \N$, subsequently:
- $\map k n := \dfrac {3^n - 1} 2$
- $\ds A_n := \bigcup_{i \mathop = 1}^{\map k n} \openint {\frac {2 i - 1} {3^n} } {\frac {2 i} {3^n} }$
Since $3^n$ is always odd, $\map k n$ is always an integer, and hence the union will always be perfectly defined.
Consider the closed interval $\closedint 0 1 \subset \R$.
Define:
- $\CC_n := \closedint 0 1 \setminus A_n$
The Cantor set $\CC$ is defined as:
- $\ds \CC = \bigcap_{n \mathop = 1}^\infty \CC_n$
From Ternary Representation
Consider the closed interval $\closedint 0 1 \subset \R$.
The Cantor set $\CC$ consists of all the points in $\closedint 0 1$ which can be expressed in base $3$ without using the digit $1$.
From Representation of Ternary Expansions, if any number has two different ternary representations, for example:
- $\dfrac 1 3 = 0.10000 \ldots = 0.02222$
then at most one of these can be written without any $1$'s in it.
Therefore this representation of points of $\CC$ is unique.
As a Limit of a Decreasing Sequence
Let $\map {I_c} \R$ denote the set of all closed real intervals.
Define the mapping $t_1: \map {I_c} \R \to \map {I_c} \R$ by:
- $\map {t_1} {\closedint a b} := \closedint a {\dfrac 1 3 \paren {a + b} }$
and similarly $t_3: \map {I_c} \R \to \map {I_c} \R$ by:
- $\map {t_3} {\closedint a b} := \closedint {\dfrac 2 3 \paren {a + b} } b$
Note in particular how:
- $\map {t_1} {\closedint a b} \subseteq \closedint a b$
- $\map {t_3} {\closedint a b} \subseteq \closedint a b$
Subsequently, define inductively:
- $S_0 := \set {\closedint 0 1}$
- $S_{n + 1} := \map {t_1} {C_n} \cup \map {t_3} {C_n}$
and put, for all $n \in \N$:
- $C_n := \ds \bigcup S_n$
Note that $C_{n + 1} \subseteq C_n$ for all $n \in \N$, so that this forms a decreasing sequence of sets.
Then the Cantor set $\CC$ is defined as its limit, that is:
- $\ds \CC := \bigcap_{n \mathop \in \N} C_n$
These definitions are all (topologically) equivalent, as shown on Equivalence of Definitions of Cantor Set.
Also defined as
Some sources define a Cantor set as any set topologically equivalent to such a set.
Also known as
Some sources refer to the Cantor set as Cantor's discontinuum.
Some sources refer to this specifically as Cantor's middle third set or Cantor's ternary set.
Also see
- Equivalence of Definitions of Cantor Set
- Fractal Dimension of Cantor Set
- Cantor Space, the Cantor set endowed with the Euclidean topology.
- Results about the Cantor set can be found here.
Source of Name
This entry was named for Georg Cantor.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Cantor set