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**1. Homework Statement**

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).