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 both of the following hold:
 * $(1): \quad Y$ is closed in $\left({X, \tau}\right)$.
 * $(2): \quad \left({Y, \preceq \restriction_Y}\right)$ is a complete lattice, where $\restriction$ denotes restriction.

Also see

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