Connectedness of R\QXQ

This one was just amazing.

Consider R\QXQ. Is it connected?

It turns out that it is polygonally path connected with at most two segments!!

And the proof is just two lines:


Consider two points p,q in R\QXQ.

Then, through p (and similarly for q) there is an infinity of lines that pass through only irrational points. Pick a line each for p and q. Call their intersection r. Then p->r->q  is the desired path.