Connected Subspace of Linearly Ordered Space

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

Let $Y \subseteq X$.

Then $Y$ is connected in $\left({X, \tau}\right)$ iff both of the following hold:
 * $(1): \quad Y$ is convex in $X$.
 * $(2): \quad \left({Y, \preceq \restriction_{Y \times Y}}\right)$ is a linear continuum, where $\restriction$ denotes restriction.

Also see

 * Subset of Real Numbers is Interval iff Connected
 * Compact Subspace of Linearly Ordered Space