Definition:Ray (Order Theory)

Definition
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:


 * $\left\{{x: x \succ a}\right\}$ (the strict upper closure of $a$), denoted $a^\succ$ or $\left({a \,.\,.\, \to}\right)$
 * $\left\{{x: x \prec a}\right\}$ (the strict lower closure of $a$), denoted $a^\prec$ or $\left({\gets \,.\,.\, a}\right)$.

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 see

 * Order Topology, a topology whose subbase consists of open rays.