Heine-Borel iff Dedekind Complete

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Theorem

Let $\struct {X, \preceq, \tau}$ be a linearly ordered space.

Then $X$ is Dedekind complete if and only if every closed, bounded subset of $X$ is compact.

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

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$Y$ is not compact

Let $\AA = \set { {\downarrow} s: s \in S}$.

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

$\blacksquare$