Is saying a set is nowhere dense the same as saying a set has no interior? I have two statements 

"A set X is nowhere dense"
"A set X has no interior"

Are these both equivalent statement?
 A: No, $\mathbb{Q}\subset\mathbb{R}$ has an empty interior, but is dense in $\mathbb{R}$.
A: For possible future use by others, it might be helpful to dissect these notions a bit more systematically in a metric space setting to see what makes them different. In particular, this will show how being nowhere dense is a super-strong way of having empty interior.
$B(x,\epsilon)$ denotes the open ball centered at $x$ with radius $\epsilon,$ where $\epsilon > 0$ is assumed.
$X$ has empty interior
This is equivalent to each of the following:
$X$ has no interior points.
For each $x$ in the space, $x$ is not an interior point of $X.$
For each $x$ in the space, there is no nonempty open set containing $x$ that is a subset of $X.$
For each $x$ in the space, there is no ball $B(x,\epsilon)$ that is a subset of $X.$
For each $x$ in the space, each ball $B(x,\epsilon)$ contains at least one point that belongs to the complement of $X.$
$X$ is nowhere dense
This is equivalent to each of the following:
$X$ is not dense in each nonempty open set.
$X$ is not dense in each ball $B(x,\epsilon).$
For each $x$ in the space, each ball $B(x,\epsilon)$ is NOT densely filled with points from $X.$
For each $x$ in the space, each ball $B(x,\epsilon)$ contains at least one nonempty open set that is a subset of the complement of $X.$
For each $x$ in the space, each ball $B(x,\epsilon)$ contains at least one ball all of whose points belong to the complement of $X.$
