\begin{definition}[Definition:Cantor Set/Limit of 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:
:$\ds \CC := \bigcap_{n \mathop \in \N} C_n$
Category:Definitions/Cantor Set
\end{definition}