Convex Set of Ordered Set is not necessarily Interval

Theorem

Let $\struct {S, \preccurlyeq}$ 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 $S$:

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

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

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $39$. Order Topology: $1$