In article <20110615233422.U18415@agora.rdrop.com>, William Elliot <email@example.com> wrote:
> Let S = QxQ as a subset of R^2. > > A rational ray is a ray from the origin O, with rational slope. > An irrational ray is a ray from O with rational slope. > > Along each rational ray, S is unbounded > and along each irrational ray, S is bounded. > > The rational rays are almost nowhere. > The irrational rays are almost everywhere. > > S is unbounded almost nowhere, bounded almost everywhere.
I don't entirely follow your point that this takes place in QxQ. A line through the origin with irrational slope can NOT pass through ANY point of QxQ, say (q1, q2), because such a point has rational slope q2/q1.
So your irrational rays lie entirely in the complement of QxQ.
In other words no point of QxQ lies on any irrational ray.
Can you try to clarify what you mean by calling an irrational ray unbounded?