Definition:Cantor Set/Limit of Decreasing Sequence

Definition
Let $I_c \left({\R}\right)$ denote the set of all closed real intervals.

Define the mapping $t_1: I_c \left({\R}\right) \to I_c \left({\R}\right)$ by:


 * $t_1 \left({\left[{a \,.\,.\, b}\right]}\right) := \left[{a \,.\,. \frac 1 3 \left({a + b}\right)}\right]$

and similarly $t_3: I_c \left({\R}\right) \to I_c \left({\R}\right)$ by:


 * $t_3 \left({\left[{a \,.\,.\, b}\right]}\right) := \left[{\frac 2 3 \left({a + b}\right) \,.\,. b}\right]$

Note in particular how:


 * $t_1 \left({\left[{a \,.\,.\, b}\right]}\right) \subseteq \left[{a \,.\,.\, b}\right]$
 * $t_3 \left({\left[{a \,.\,.\, b}\right]}\right) \subseteq \left[{a \,.\,.\, b}\right]$

Subsequently, define inductively:


 * $S_0 := \left\{{\left[{0 \,.\,.\, 1}\right]}\right\}$
 * $S_{n+1} := t_1 \left({C_n}\right) \cup t_3 \left({C_n}\right)$

and put, for all $n \in \N$:


 * $C_n := \bigcup S_n$

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 is defined as its limit, i.e.:


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