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:


 * $\left\{{x: x > a}\right\}$, denoted $\left({a, \infty}\right)$ or occasionally $\left]{a, \infty}\right[$
 * $\left\{{x: x < a}\right\}$, denoted $\left({-\infty, a}\right)$ or occasionally $\left]{-\infty, a}\right[$

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


 * $\left\{{x: x \ge a}\right\}$, denoted $\left[{a, \infty}\right)$ or occasionally $\left[{a, \infty}\right[$
 * $\left\{{x: x \le a}\right\}$, denoted $\left({-\infty, a}\right]$ or occasionally $\left]{-\infty, a}\right]$