Suppose that $f: \mathbb R^n \to \mathbb R^n$ is a bijection and $n\geq2$. Can $f$ send every open set onto non-open set?
I do not know what exactly to write about this problem. Did I try anything? No, I do not have a good idea. Why am I asking this? Because here is this very highly upvoted question of Willie Wong that I think of, and, it seems to me as a good start to investigate what exactly can bijections "do" and what they can´t, so, as a starting point, I decided to ask this question.
Edit: Thomas wrote a useful comment that $\emptyset$ and $\mathbb R^n$ are mapped onto open sets. So to exclude trivialities, suppose that from consideration we exclude the empty set and the whole space $\mathbb R^n$, to make this more interesting.