\begin{definition}[Definition:Ray (Order Theory)/Open]
Let $\left({S, \preccurlyeq}\right)$ be a totally ordered set.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
Let $a \in S$ be any point in $S$.
The following sets are called '''open rays''' or '''open half-lines''':
: $\left\{{x \in S: a \prec x}\right\}$ (the strict upper closure of $a$), denoted $a^\succ$
: $\left\{{x \in S: x \prec a}\right\}$ (the strict lower closure of $a$), denoted $a^\prec$.
\end{definition}