1. The problem statement, all variables and given/known data 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? 3. The attempt at a solution I considered it to be yes. Since in the ordinary situation, the assertion is obviously valid: check out the real line or the complex plane with dictionary order, in case of Y is not empty. But I wonder if it holds when Y is an empty set or a set with only a single point. And besides, I'm not quite sure with my own judgement, since I didn't think of any special situation (if any).