 Solution

The set is a rational comb with a connecting spine: it contains the whole vertical segment $x=0$, and every horizontal segment at rational height. Any point with $x>0$ has rational height and can move horizontally to the vertical segment without leaving the set. It can then move vertically to $(0,0)$. Points already on the vertical segment use only the second step. Concatenating these polygonal paths connects any two points. ConsequentlyNo claim about a horizontal segment at irrational height is needed: at such a height, the only point in the set lies on the connecting spine.