I've tried to think this, help would be appreciated to get me going a gain.

If we have two nonempty open set $\displaystyle A,B \subset {R}^{n}$ so that $\displaystyle A \cap B = \emptyset$(two separate sets). How could I show that complement of union A and B is also nonempty, $\displaystyle \complement (A \cup B) \neq \emptyset$.

I'm thinking something like, because the complement is $\displaystyle {R}^{n} \setminus (A \cup B)$ so because its R^n there must be some points that belong to that complement. But basicly I think that don't work (just to say because it's R^n) because I have show that it (complement of the union is nonempty) follows from the notion that A,B are open nonempty sets.

Any ideas and help? Thanks!