Hausdorff distance in Cantor set

Question

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Let $C_n$ denotes the nth stage in the construction of the Cantor ternary set, i.e. $C_0=[0,1]$, $C_1=[0,1/3] \cap [2/3,1]$ and so on. Find the Hausdorff distance between $C_n$ and $C_{n+1}$

Answer 1

First of all $C_{n}\subset C_{n-1}$ and therefore: $$ d_H(C_n,C_{n-1})=\sup_{y\in C_{n-1}}\inf_{x\in C_{n}}d(x,y). $$ Now for all $y\in C_{n}$, we have $d(x,y)=0$. However for all $y\in C_{n-1}\cap C_n^c$, we have non-zero distance. The supremum of the distance is obtained when we choose $x$ the middle point in the excluded middle intervals of $C_{n-1}$ which is the length of each interval in $C_{n}$. This length is $\frac 1{3^n}$ and therefore:

$$ d_H(C_n,C_{n-1})=\sup_{y\in C_{n-1}}\inf_{x\in C_{n}}d(x,y)=\frac 1{2\times 3^n} $$