Show that $S=\mathbb R^2\setminus\{(a,b):a,b\in\mathbb Q\}$ is path connected. > Show that $S=\mathbb R^2\setminus\\{(a,b):a,b\in\mathbb Q\\}$ is path connected. By definition of path connected, there should exist continuous mapping $f:[0,1]\rightarrow \mathbb R^2$ s.t. $f(0)=a,f(1)=b$ for all $a,b\in\mathbb R^2$. I have utterly no idea how to find such map. Please help.

This result can be generalized to an arbitrary countable set $A \subset \mathbb{R}^2$. Since $A = \mathbb{Q}^2$ is countable, and there exist uncountably many lines in $\mathbb{R}^2$ through the point $x$, there must exist a line $\ell_1$ through $x$ that does not intersect the set $A$. Analogously, there exists a line $\ell_2$ through $y$ such that $\ell_2$ does not intersect $A$. Let $\ell_1$ and $\ell_2$ intersect at $p \in \mathbb{R}^2 - A$ and choose the path $x\to p \to y$.