Find the interior of the set $\{ 1, 1/2, 1/3,\ldots\}$. If we assume that this set is subset of $\mathbb{R}$ then the interior set is the empty set. 

If you define the interior as: 
$$\text{int }S :=\{s\in S\ |\ \ \forall \ \varepsilon >0, S\cap B(s,\varepsilon)\backslash\{s\}\neq \emptyset\}$$
...then the interior is empty. In other words, any element of your set is going to have a ball around it which when we look at, has nothing in $S$ other than the point you started with.
For some arbitrary point $s=\frac{1}{n}$ in $S$, think about balls around $s$ of size less than $\frac{1}{n}-\frac{1}{n+1}$.
(In my notation, $B(s,\varepsilon)$ deonotes the open $\varepsilon$-ball around $s$)
