As the title says I was wondering (being vaguely inspired by a question from Hatcher asking about the fundamental group of the first space) whether $\Bbb R^2\setminus \Bbb Q^2$ and $\Bbb R^2\setminus \Bbb Q^2\cup \{(0,0)\}$ are homeomorphic.
My gut feeling is that they are, they have the same properties as far as connectedness, compactness and separation axioms are concerned, supporting this feeling, but I haven't been able to prove (or disprove) this fact.