Non-end points of Cantor ternary set I have studied the Cantor Ternary Set $C$ . We know that the Cantor ternary  set is uncountable. Take the set $A$ as the set of all end points of the open removed segments. That is, $A=\{0,\frac{1}{3},\frac{2 }{3},\frac{1}{9},\frac{2}{9}....\}$. Now the set $C\setminus A$ is still uncountable. So the cardinality of $C$ was entirely determined by the non-end points.
My question is: Can someone write any $5$ (at least) members of $C\setminus A$?
I know $\frac{1}{4}, \frac{3}{10} \in C\setminus A$ but I don't know about any others.
Any help will be appreciated!
 A: Any element $Z$ in the Cantor Set can be written as :
$Z= \sum_{n=1}^\infty  x_n/3^n$ where $x_n=0$ or $2$.
You can substitute any combination of $0,2$ and have your number belonging to the 
cantor set
A: Any real number between $0$ and $1$ whose ternary expansion doesn't have any $1$s is in the Cantor set. So, for example, each of the following ternary expansions corresponds to a point in the Cantor set:


*

*$0.2020202020...$

*$0.220022002200...$

*$0.202200222000...$
and so forth. It's easy to check that these don't correspond to endpoints of the Cantor set.
A: Noah Schweber's answer gives a complete characterization of points in the Cantor set. A point is an endpoint if it eventually ends in all 0s or all 2s. Otherwise, it is a non-endpoint.
I'll just add a couple other ideas:
The Cantor set has some nice symmetries, like $x\mapsto \frac13 x$ and $x\mapsto 1-x$. These linear symmetries preserve endpoint-ness. So if you know that $\frac14$ is a non-endpoint, then you also know that $\frac{3}{4}$ and $\frac{1}{12}$ and $\frac{11}{12}$ and so on are non-endpoints. Similarly, $\frac{3}{10}$ gets you $\frac{1}{10},\frac{7}{10},\frac{9}{10},\frac{1}{30},$ and so on.
Finally, as Noah's third example illustrates, you can use ternary expansions to describe concrete irrational elements of $C\setminus A$ as well. For another example, we have $\theta\left(\frac13\right)-1=0.2002000020000002\ldots_3\in C\setminus A$. See my answer here for the meaning of the $\theta$ notation: Which irrationals are contained in the Cantor set?
