 Solution

Use the continuous slope map $(x,y)\mapsto x/y$ on the given set, where $y>0$. The attained ratios are positive rational numbers. Choosing the irrational cut $\sqrt{2}$, the two subsetsare disjoint, relatively open, and cover the entire set. Both are nonempty: ratios one and two are attained inside the square. This irrational-cut separation of rational slopes is a separation, soThe individual rational-slope rays are connected, but their common origin is excluded; their separate connectivity does not imply connectivity of their union.