Convex Set Characterization (Order Theory)

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

Let $C \subseteq S$.

$(2)$ implies $(1)$
Follows from Upper Set is Convex, Lower Set is Convex, and Intersection of Convex Sets is Convex Set (Order Theory).

$(1)$ implies $(3)$
Let $C$ be a convex set in $S$.

Let $U$ and $L$ be the upper and lower closures of $C$, respectively.

Since $C \subseteq U$ and $C \subseteq L$:
 * $C \subseteq U \cap L$.

Let $p \in U \cap L$.

Then $a \preceq p \preceq b$ for some $a, b \in C$.

Since $C$ is convex, $p \in C$.

Since this holds for all $p \in U \cap L$:
 * $U \cap L \subseteq C$.

Since we know that $C \subseteq U \cap L$, $C = U \cap L$.

$(3)$ implies $(2)$
Follows from Upper Closure is Upper Set and Lower Closure is Lower Set.