Existence of a dense proper subset with non-empty interior? 
*

*What is an example of a topological (or particularly a metric) space $X$ having a proper non-open dense subset with non-empty interior?

*Is it possible for a closed dense subset of a topological space to have an empty interior?

*Finally: on a similar note: I've noticed that $\mathbb{Q}$ doesn't work, why?


(I will post my solution below as a Q& A type problem/solution:)
 A: 
Example of a closed dense subset with non-empty interior 
As a counter example of a proper dense subset of a set of topological space with non-empty interior consider 
\begin{equation}
D:=(-\infty,0]\cup (\mathbb{Q}\cap(0,1)) \cup[1,\infty)
\end{equation}
 in $\mathbb{R}$ with its usual topology, then since: 


*

*$(-\infty,0]$ and $\cup[1,\infty)$ are closed in $\mathbb{R}$, they must equal to their closure.

*$\mathbb{Q}\cap(0,1)$ is not open and dense in $[0,1]$ since: 
$\mathbb{Q}$ is not open and is density $\mathbb{R}$, taking intersections with $(0,1)$ yields the conclusion.  


therefore we can conclude that $D$ is not closed, dense in $\mathbb{R}$ and a proper subset thereof.  
Moreover, the subset $(\infty,0)\cup(0,\infty)$ is open in $D$ (and infact is the largest such subset, therefore is equal to its interior), hence the interior of $D$ is non-empty.  

Concerning the existence of a closed dense subset with empty interior:
By the definition of density the closure of $D$ must be equal to $X$.  However the axioms of a topological space assure that $X$ is closed in $X$ and by the since $D$ was assuemd to be closed also it must be equal to its closure, therefore $X=D$ if $D$ is closed and dense in $X$.  
There is only one possibility of a topological space  $X$ which has a closed dense subset $D$ with empty interior, that is $X$ itself must be the empty set.
At this point it becomes a question of convention!  Most people require that the initial object in the category of topological spaces be the one-point space, therefore the empty set (though it does satisfy the axioms of a topology using the empty-topology thereon) is not generally regarded as begin a topological space.  Hence under this convention there is no such space (if you reject this very standard convention (and almost definition at this point) then our non-mainstream example would be the only one which is the the empty topological space with the topology $\{ \emptyset \}$).  (on a personal note, I require topological space to be non-empty).  

Concerning the empty interior of $\mathbb{Q}$: 
The empty interior follows from $\mathbb{R}-\mathbb{Q}$ is dense in $\mathbb{R}$ and remarked is not from $\mathbb{Q}$'s density in $\mathbb{R}$.  
The reason being that in every open ball around a rational number there exists and irrational number, hence there cannot be an open ball in $\mathbb{R}$ properly contained in $\mathbb{Q}$.  Since the open balls generate the metric topology on $\mathbb{R}$, then there cannot be a (non-empty) open subset of $\mathbb{Q}$ contained within $\mathbb{Q}$.  Now using the definition of interior as the union of all open subsets of a set, this implies the interior of $\mathbb{Q}$ is empty. 
