Are points in the cantor set limits of endpoints

Question

If $x\in C$ the cantor set and $S$ is the set of all endpoints in the iterative construction (i.e. $C=\cap C_i$ and $C_i= [a_1^i,b_1^i]\cup...\cup[a^i_{2^i},b_{2^i}^i]$ then $S =\cup_i\cup_k \{a^i_k,b^i_k\}$), is $x$ a limit point of $S$? Intuitively it makes sense to me that they are: firstly the set of all sequences of $S$ is uncountable so that is a sanity check, secondly in the picture of $C_i$ we see that the intervals are getting smaller and smaller around points.

Answer 1

If $x$ belongs to the Cantor set then it is at distance of atmost $\frac 1 3$ from one of points $\frac 1 3, \frac 2 3$ in the first step of the construction of $C$. It is at distance of atmost $\frac 1 {3^{2}}$ from one of end points in the second step of the construction of $C$, and so on. So YES, $x$ is a limit point of the set of all end points.

Answer 2

In the homeomorphic model $\{0,1\}^{\Bbb N}$ of the Cantor set (discrete two point spaces in a countable product), the endpoints of the construction correspond to the points that are eventually $0$. And that set is clearly dense in the product topology and so are the endpoints in the Cantor set.