Connected Subspace of Linearly Ordered Space

Theorem
Let $\left({X, \preceq}\right)$ be a totally ordered set, equipped with its order topology so that it is considered as a topological space.

Then a topological subspace $Y \subseteq X$ is connected 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

 * Only Intervals are Connected
 * Compact Subspace of Totally Ordered Set