= 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. Consequently
$$
\boxed{\text{the subspace is path-connected, hence connected}.}
$$
No 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.
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