is $(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}))\cup((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q})$ connected? path connected?

Question

let $$X=(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}) ) \cup ((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q}) $$ and let $$\tau=\tau (\text{euclid})$$ what are the connected components of $$(X,\tau)$$ what are the path connected components?

Answer 1

It is not path connected and the components are the points. There are no path between two points.

Take for example $A(1,\pi)$ and $B(\pi,1)$. Any path from $A$ to $B$ must cross the line $y=x$, which has no point in $X$.

For any other two points, simply make an appropiate translation.