Heine-Borel iff Dedekind Complete

Theorem
Let $(X, \preceq, \tau)$ be a linearly ordered space.

Then $X$ is Dedekind Complete iff for each nonempty subset $Y$ of $X$:
 * $Y$ is compact iff $Y$ is closed and bounded in $X$.

Proof
The forward implication follows from Heine-Borel Theorem/Dedekind-Complete Space.

Suppose that $X$ is not Dedekind complete.

Then $X$ has a non-empty subset $S$ with an upper bound $b$ in $X$ but no supremum in $X$.

Let $a \in S$ and let $Y = {\bar\downarrow}S \cap {\bar\uparrow} a$.

$Y$ is nonempty and bounded below by $a$ and above by $b$.

$Y$ is closed in $X$
Let $x \in X \setminus Y$.

Then $x \prec a$ or $x$ strictly succeeds every element of $S$.

If $x \prec a$, then $x \in {\downarrow}a \subseteq X \setminus Y$.

If $x$ strictly succeeds each element of $S$, then it is an upper bound of $S$.

Since $S$ has no supremum in $X$, it has an upper bound $p \prec x$.

Then $x \in {\uparrow p} \subseteq X \setminus Y$.

$Y$ is not compact
Let $\mathcal A = \{{ {\downarrow}s: s \in S }\}$.

Then $\mathcal A$ is an open cover of $Y$ with no finite subcover.