Definition:Ray (Order Theory)/Closed

Definition
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:


 * $\left\{{x: x \succcurlyeq a}\right\}$ (the upper closure of $a$), denoted $a^\succcurlyeq$ or $\left[{a \,.\,.\, \to}\right)$
 * $\left\{{x: x \preccurlyeq a}\right\}$ (the lower closure of $a$), denoted $a^\preccurlyeq$ or $\left({\gets \,.\,.\, a}\right]$.

Also known as
A closed ray is also sometimes referred to as a closed half-line.

The notations:
 * $\left[{a \,.\,.\, \to}\right)$ for $a^\succcurlyeq$
 * $\left({\gets \,.\,.\, a}\right]$ for $a^\preccurlyeq$

can also be used.