Compact Subspace of Linearly Ordered Space/Lemma 1

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

Let $Y \subseteq X$.

Then $Y$ is compact in $\left({X, \tau}\right)$ iff all of the following hold:
 * $(1): \quad Y$ has a greatest element and a smallest element.
 * $(2): \quad \left({Y, \preceq \restriction_{Y \times Y}}\right)$ is Dedekind complete, where $\restriction$ denotes restriction.
 * $(3): \quad Y$ is closed in $X$.

Also see

 * Heine–Borel Theorem: Special Case
 * Connected Subspace of Linearly Ordered Space