I've been stuck with this question for a couple days so I was hoping to get some answers here. My question is about subsets in $\Bbb R^2$ with the usual topology.
I know that $\Bbb R^2 \setminus \Bbb Q^2$ is connected because it is path connected, which is easy enough to see. Unless I'm mistaken, $\Bbb R^2 \setminus \Bbb I^2$ (where $\Bbb I = \Bbb R \setminus \Bbb Q$ is the set of irrational numbers) is also path connected.
My question is if $(\Bbb Q \times \Bbb I)\cup(\Bbb I \times \Bbb Q) = (\Bbb R^2 \setminus \Bbb Q^2) \cap (\Bbb R^2 \setminus \Bbb I^2)$ is also connected. If it is path connected, it is tougher to see than in the case of the two previously mentioned sets, but I'm only looking to see if it is connected, not path connected, I mention path connectedness because in the previous sets that was a quick way to show they are indeed connected. I've been trying to find clopen sets but it's also proving difficult, be it because there aren't any or because I can't find them.
I would greatly appreciate some pointers, thank you!
