Definition:Ray (Order Theory)

Definition
Let $\left({ S, \le }\right)$ be a totally ordered set.

Let $<$ be the reflexive reduction of $\le$.

Then for any point $a \in S$, the following sets are called open rays or open half-lines:


 * $\left\{{x: x > a}\right\}$ (the strict up-set of $a$), denoted ${\dot\uparrow} a$ or $(a \,.\,.\, \to)$
 * $\left\{{x: x < a}\right\}$ (the strict down-set of $a$), denoted ${\dot\downarrow} a$ or $(\gets \,.\,.\, a)$.

The following sets are called closed rays or closed half-lines:


 * $\left\{{x: x \ge a}\right\}$ (the upper closure of $a$), denoted ${\bar\uparrow} a$ or $[a \,.\,.\, \to)$
 * $\left\{{x: x \le a}\right\}$ (the lower closure of $a$), denoted ${\bar\downarrow} a$ or $(\gets \,.\,.\, a]$.

Also see

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