The set of numbers whose decimal expansions contain only 4 and 7 Let $S$ be the set of numbers in $X=[0,1]$ that when expanded as a decimal form, the numbers are 4 or 7 only.
The following are the problems.
a), Is S countable ?
b), Is it dense in $X$ ?
c), Is it compact ?
d), Is it perfect ?
For a), I want to say that it is intuitively, but I have no idea how to prove this. I tried to come up with a bijection between $S$ and $\Bbb Z$ but I couldn't find one.
For b), my understanding of a set being "dense" means that all points in $X$ is either a limit point or a point in $S$. Am I right? Even if I were, I am not sure how to show this.
For c), my intuition tells me that it is because it is bounded. So if I could show that it is closed I will be done, I think.  But I am still iffy with the idea of limit points, and I am not sure what kind of limit points there are in $S$.
For d), Because I can't show that it's closed I am completely stuck.  
I am teaching myself analysis, and I only know up to abstract algebra. Since I never took topology, please give me an explanation that helps without knowledge of advanced math.

Hints:  For b, can you get close to $0.2?$
For c, you are correct that it is bounded, so you need to investigate closed.  Let $y \in [0,1]$, but $y \not \in S$.  Then there is some digit of $y$ that is not $4$ or $7$....
For d, you need to show that any point of $S$ is a limit of a sequence of other elements of $S$.  Let $x \in S$.  Can you find a list of other elements in $S$ that get closer and closer to $x$?  For starters, can you describe another point within $\pm 0.1$ of $x$?
