Connectedness of coordinates with one rational point

[hypermonkey2]

Hi all, i found this problem in a topology book, but it seems to be of an analysis flavour. I'm stumped.

Show that the collection of all points in R^2 such that at least one of the coordinated is rational is connected.

My gut says that it should be path-connected too (thus connected), but I am finding the proof elusive... any thoughts?

cheers

 

[Citan Uzuki]

It is path-connected. Try paths consisting of horizontal and vertical segments moving along straight lines. For instance, to move from (0, √2) to (π, 1/2), you could first move along the straight line segment from (0, √2) to (0, 1/2), and then along the straight line segment from (0, 1/2) to (π, 1/2). Now find a way to generalize that line of thought.