Is it true that $F\cap \overset{\circ}{E}=\emptyset\implies F\cap \overset{\_}{E}=\emptyset$? Let $E,F\subset \mathbb{R}^n$ where $F$ is open and $E$ is arbitrary. Is it true that:
$$
F\cap \overset{\circ}{E}=\emptyset\implies F\cap \overset{\_}{E}=\emptyset
$$
Intuitively I think this is true, but my intuition on open and closed sets has proven to be not that good. How to prove it or how to construct a counter example?
Edit: How about 
$$
F\cap E=\emptyset\implies F\cap \overset{\_}{E}=\emptyset
$$

Not at all, since the frontier of a set $S$ in a topological space is precisely 
$$\partial S=\bar S\setminus\mathring S\subset \bar S$$
and $\partial S\cap \mathring S=\varnothing$.
