Let \(X\) be an ordered set. If \(Y\) is a proper subset of \(X\) that is convex in \(X\), does it follow that \(Y\) is an interval or a ray in \(X\)?
Solution
No. Take \(X=\mathbb Q\) with its usual order and \[ Y=\{q\in\mathbb Q:-\sqrt2<q<\sqrt2\}. \] This is convex in \(\mathbb Q\): if \(p<q<r\) with \(p,r\in Y\), then necessarily \(q\in Y\). It is a proper subset of \(\mathbb Q\).
However, \(Y\) is not an interval with endpoints in \(\mathbb Q\), since its two cuts are at the irrational numbers \(\pm\sqrt2\), and it is not a ray since it is bounded on both sides in \(\mathbb Q\). Thus a proper convex subset of an ordered set need not be one of the intervals or rays whose endpoints belong to the ambient ordered set.