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.