The definition of dense sets

Question

I am confused with the definition of dense sets. By definition, a subset $S$ of a metric space $X$ is dense in $X$ if the closure of $S$ in $X$ is equal to $X$. By definition, this means that every point of $X$ is either in $S$ or a limit point of $S$. My question is that does this mean every point of $X$ is a limit point of $S$? This is the explanation I often see about the subset $S$ being dense in $X$. But if an arbitrary point of $X$ actually lies in $S$, how do we know it is a limit point of $S$? By definition, for a point $x$ of $S$ to be a limit point of $S$, every epsilon ball around $x$ has to contain a point of $S$ which is different from $x$. And I am not sure why this will hold for any metric space $S$.

I am aware that the definition of a point being a limit point of a set is different from the definition of a point being the limit of a sequence in that set. With the latter definition, for any point $x$ of a set $S$, we can just take the constant sequence $(x_n)=x$ and say $x$ is the limit of this sequence. But this is different from $x$ being a limit point of $S$. If someone knows the answer, please let me know.

Answer 1

Let $X=\{0,1\}$, with its usual topology. Then $X$ is a dense subset of itself. However, neither $0$ nor $1$ is a limit point. So,no, $S$ being a dense subset of $X$ does not mean that every point of $X$ is a limit point of $S$.

Answer 2

"My question is that does this mean every point of X is a limit point of S?"

Not unless every point of $S$ is a limit point of $S$. So to get a counter example take any set of $S$ which has points that aren't limit points of $S$.

A point $x\in S$ that is not a limit point. Is such that there exists a value $r$ so that for any point $y \in S; y \ne x$ then $d(x,y) \ge r$. In other words $x$ is a singleton.

So if $S$ has singletons that is a counter example.