Open Ray is Dual to Open Ray

Theorem
Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $R$ be an open ray in $\left({S, \preceq}\right)$.

Then $R$ is an open ray open ray in $\left({S, \succeq}\right)$, where $\succeq$ is the dual ordering of $\preceq$.

Proof
By the definition of open ray, there is some $p \in S$ such that:


 * $R$ is the strict upper or strict lower closure of $p$ with respect to $\preceq$

By Strict Lower Closure is Dual to Strict Upper Closure, the dual statement is:


 * $R$ is the strict upper or strict lower closure of $p$ with respect to $\succeq$

Thus $R$ is an open ray in $\left({S, \succeq}\right)$.