Open Ray is Dual to Open Ray
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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 $\struct {S, \succeq}$.
$\blacksquare$