Definition:Ray (Order Theory)/Open

Definition
Let $\struct {S, \preccurlyeq}$ 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:


 * $\set {x \in S: a \prec x}$ (the strict upper closure of $a$), denoted $a^\succ$
 * $\set {x \in S: x \prec a}$ (the strict lower closure of $a$), denoted $a^\prec$.

Also known as
An open ray is also sometimes referred to as an open half-line.

The notations:
 * $\openint a \to$ for $a^\succ$
 * $\openint \gets a$ for $a^\prec$

can also be used.

Also see

 * Definition:Closed Ray


 * Definition:Upward-Pointing Ray
 * Definition:Downward-Pointing Ray


 * Definition:Order Topology: a topology whose sub-basis consists of open rays.