I know the following definition : a set $A$ is dense in a metric space $X$ if every point of $X$ is a closure point of $A$. That is, for every $x\in X$, every neighborhood of $x$ contains a point of $A$.
However, a book I read said this :
A subset $S$ of $\Bbb R$ is dense if every point of $\Bbb R$ is a limit point (cluster point) of $S$ i.e. $S'=\Bbb R$.
This confused me. I know that $\overline D=X$ does not necessarily mean that $D'=X$. For example, let $X=[1,2]\cup${$3$} and $D=(1,2)\cup${$3$}. Then $\overline D=X$ but $D'=[1,2]≠X$. According to the definition I know $D$ is dense in $X$. But if I apply the latter definition (taking $X$ in place of $\Bbb R$) then it is not dense in $X$.
If every point of $\mathbb R$ is a limit point of $S$ then indeed $S$ is a dense subset of $\mathbb R$.
But the opposite of this is not true in general (so do not read it as a definition).
If e.g. $\mathbb R$ is equipped with discrete topology and $S=\mathbb R$ then $S$ is a dense subset of $\mathbb R$.
However, in this case $S'=\varnothing$.