Are $\Bbb R^2\setminus \Bbb Q^2$ and $\Bbb R^2\setminus \Bbb Q^2\cup \{(0,0)\}$ homeomorphic?

Solution 1:

It is a theorem due to Brouwer (1913) that for any two dense countable subsets $A, B\subset R^n$, there is a homeomorphism $R^n\to R^n$ sending $A$ to $B$ bijectively. See also "General Topology" by Engelking (he has this as an exercise 4.5.2, with a detailed hint). If I remember it correctly, Hirsch in "Differential Topology" also has this as an exercise where instead of a homeomorphism he asks for a diffeomorphism. Lastly,

M. Morayne, Measure preserving analytic diffeomorphisms of countable dense sets in $C^n$ and $R^n$, Colloq. Math. 52 (1987), no. 1, 93–98.

proves that for any two countable dense subsets $A, B$ in $R^n$, $n\ge 2$, there exists an analytic volume-preserving diffeomorphism of $R^n\to R^n$ sending $A$ to $B$ bijectively.

So, the conclusion is that your spaces are homeomorphic.