Convex Subsets of Ordered Sets: Interval or Ray in Topology?

[Ka Yan]

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?

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).

 

[morphism]

Start by looking at the definitions of "interval" and "ray" in X. Then look at the definition of Y being convex in X; does this imply it's an interval or a ray?

 

[HallsofIvy]

The "X is an ordered set" is important here! That should certainly be considered in the definition of "interval" and "ray".