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