Open Ray is Dual to Open Ray

Theorem

Let $\struct {S, \preceq}$ be a totally ordered set.

Let $R$ be an open ray in $\struct {S, \preceq}$.

Then $R$ is an open ray in $\struct {S, \succeq}$, 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 $\struct {S, \succeq}$.

$\blacksquare$