# Dense sets and Empty Interior

if $A$ is dense in $X$, is there a relation which shows in which cases $A$ has empty interior ? $\mathbb{Q}$ has an empty interior as a dense set in $\mathbb{R}$, so does its complementary in $\mathbb{R}$. The open interval $(a,b)$ is dense in the closure of the same interval but does not have an empty interior. The complementery of $(a,b)$ in its closure consists of the two points $a$ and $b$, which has empty interior. So is there a general statement relating the fact of being dense and having empty interior ?

• Do you know that $X\setminus \overset{\Large\circ}{M} = \overline{X\setminus M}$ for all $M\subset X$? Commented Apr 27, 2015 at 15:28
• It does not work if X=[a,b] and M=(a,b) and i dont see how the dense set is then related to the fact of having empty interior.
– ivo
Commented Apr 27, 2015 at 16:12
• "A subset of a topological space has empty interior if and only if its complement is dense." The denseness (or not) of a subset is tied to the emptiness (or not) of the interior of its complement. Commented Apr 27, 2015 at 17:25