Solution
= Solution
Let $D$ be this union of rational-coordinate horizontal and vertical <lines>. A point $(x,y)\in D$ has either rational $x$, in which case it moves vertically to $(x,0)$ then horizontally to $(0,0)$, or rational $y$, in which case it moves horizontally to $(0,y)$ then vertically to $(0,0)$. Every segment remains on a <line> with a rational fixed coordinate.
Thus \b[$D$ is <path-connected> and <connected>]. It is not necessary for both coordinates to be rational.