Showing that $\mathbb{C}$\ {$x+iy|x,y\in \mathbb{Q}$} is connected

Question

This I find it really hard to solve. I suppose the set {$x+iy|x,y\in \mathbb{Q}$} is neither closed or open. But I just cannot seem to find a way to go forward. Can someone help me out. Thanks

Answer 1

This is a kind of dual to a question asked just yesterday. The solution is similar: every point in your set is path-connected to the point $\sqrt 2+\sqrt 2i$, so every point is path-connected to every other point.

Indeed, if $a \notin \mathbb Q$, then we have a path $a+bi \to a+\sqrt 2i \to \sqrt 2+\sqrt 2i$; the case $b \notin \mathbb Q$ is similar.

(I suppose I shouldn't say that two points are "path-connected", because that adjective properly applies to the whole set; but you know what I mean.)

Answer 2

Hint 1: Find a path connecting 2 points with both coordinates irrational.

Hint 2: Connect all points with one coordinate irrational to a point with both coordinates irrational.