Exercise of spaces path connected

Question

The exercise says: Let $X$ be a subspace of $\mathbb{R}^2$ given by points with at least one irrational coordinate. Show that $X$ is path connected.

Answer 1

Hint: Let $\pi(x,y)=y$. How connected is $\pi\bigl((\mathbb R\setminus\mathbb Q)\times\mathbb Q\bigr)$?

Answer 2

Of all lines through $a\in X$, only countably hit a rational point. Hence for given $a,b\in X$ we can certainly find non-parallel $X$-avoiding lines through them.

Answer 3

Let's fix an element in $X$. I am going to choose a point with both coordinates being irrational, for instance $(w,w)\in X$ with $w=\sqrt{2}$.

To prove that $X$ is path connected is enough to prove that for any point $(a,b)\in X$ we can find a path inside $X$ connecting $(a,b)$ to $(w,w)$. Let's prove it

Let $(a,b)\in X$. WLOG say $a$ is irrational. Then go from $(a,b)$ to $(a,w)$ vertically and then go from $(a,w)$ to $(w,w)$ horizontally. Note that this is a path inside $X$ connecting $(a,b)$ and $(w,w)$.

So we are done.