Let $x \in [0,1]$ and consider the set of all ternary expansions in the interval $[0,1]$ and in the Cantor Set whose first n digits are equal to those of ternary expansion of $x$.
I believe that for some $x = 0.a_1a_2...a_n...$, this set can be denoted as
$$ A_n = \{c = \sum^n_{i=1} \frac{a_i}{3^i} + \sum^{\infty}_{j=n+1} \frac{b_j}{3^j}| b_j \in \{0,2\}\} $$
where $a_i$ is the $i^{th}$ digit of the ternary expansion of $x$.
I would like to know whether this set is open and or closed under the standard topology of R, and whether a similar approach can be done for all expansions in general, not just ternary expansions.
Thanks