$X = \{0 ,1\}^{\mathbb N}$ be the metric space. Can anyone please tell me how to define a continuous injective function from $X = \{0 ,1\}^{\mathbb N}$ to the cantor set ?
Can anyone please give an idea ?
general-topologyreal-analysis
$X = \{0 ,1\}^{\mathbb N}$ be the metric space. Can anyone please tell me how to define a continuous injective function from $X = \{0 ,1\}^{\mathbb N}$ to the cantor set ?
Can anyone please give an idea ?
Best Answer
For each $x\in X$, we have a sequence of zeros and ones. Meanwhile the Cantor set is the set of all real numbers in the unit interval whose ternary expansion contains no $1$'s. So the natural map would be to send a given sequence $x=(x_n)$ of zeros and ones to the number in the Cantor set whose ternary expression is $\sum a_n/3^n$, where $a_n=\begin{cases} 0 , x_n=0\\2, x_n=1\end{cases}$.
The map is automatically continuous because the $X$ is discrete.
It remains to prove injectivity. But that's straight forward, because different sequences result in numbers with different ternary representations.