= 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 $\sqrt2$, the two subsets
$$
A=\{x/y<\sqrt2\},\qquad B=\{x/y>\sqrt2\}
$$
are 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, so
$$
\boxed{\text{the subspace is not connected}.}
$$
The individual rational-slope rays are connected, but their common origin is excluded; their separate connectivity does not imply connectivity of their union.
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