Order Topology is Hausdorff

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

Then $\left({X,\tau}\right)$ is a Hausdorff space.

Proof
Let $x, y \in X$ with $x \ne y$.

Since $\le$ is a total ordering, either $x \prec y$ or $y \prec x$. Assume WLOG that $x \prec y$.

If there is a $z \in X$ such that $x \prec z \prec y$, then ${\downarrow}z$ and ${\uparrow}z$ separate $x$ and $y$.

Otherwise, by Mind the Gap, ${\downarrow}y$ and $\uparrow x$ separate $x$ and $y$.

Since any two distinct points can be separated by neighborhoods, $\left({X,\tau}\right)$ is a Hausdorff space.