Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point

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

Let $A \subseteq S$ be a connected space.

Let $p \in A$ be a point of $A$ which is not an endpoint of $A$.

Then $p$ is a cut point of $A$.

Proof
We have that $A \setminus \left\{ {p}\right\}$ is separated by $\left\{ {x \in A: x \prec p}\right\}$ and $\left\{ {x \in A: p \prec x}\right\}$.

If $p$ is an endpoint of $A$, then either:
 * $\left\{ {x \in A: x \prec p}\right\} = \varnothing$

or:
 * $\left\{ {x \in A: p \prec x}\right\} = \varnothing$