Convex Set of Ordered Set is not necessarily Interval

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Theorem

Let $\left({S, \preccurlyeq}\right)$ be an ordered set.

Let $C$ be a convex set of $S$.


Then it is not necessarily the case that $C$ is an interval of $S$.


Proof

Consider the open ray of $S$:

$R = \left\{ {x \in S: a \prec x}\right\}$

for some $a \in S$.

From Ray is Convex, $R$ is a convex set of $S$.

But $R$ is not an interval of $S$.

$\blacksquare$


Sources