Is $\Bbb R^2 -\Bbb Q^2$ connected? I need to prove whether or not $X=\Bbb R^2 -\Bbb Q^2$ is connected. The definition of connectedness I am using is a space X is connected if it is not the union of two disjoint nonempty open sets. This is a very new concept for me so I am not exactly sure how to set it up. I believe we will prove it by contradiction, but I do not know where to start.

Take two points $x,y \in X$. Since $\Bbb{Q}^2$ is countable and the number of lines going through $x$ is uncountable (in bijection with $[0,\pi)$), there are uncountably many lines going through $x$ and contained in $X$. The same applies to $y$. Therefore, you can find lines going through $x$ and $y$ respectivly that are not parralel, and thus intersect each other. This proves that $X$ is connected. 
