 Irrational-cut separation of rational slopes

In a plane set with $x,y>0$ and rational ratio $x/y$, any irrational number $a$ lying between two attained ratios separates the set into the nonempty relatively open sets $x/y<a$ and $x/y>a$. This exhibits a failure of connectedness using the continuous slope map, even though each individual ray is path-connected.