Definition:Cantor Set/Limit of Decreasing Sequence

Definition
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 $n \in \N$:


 * $C_n := \displaystyle \bigcup S_n$

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 $\mathcal C$ is defined as its limit, that is:


 * $\mathcal C := \displaystyle \bigcap_{n \mathop \in \N} C_n$