# Definition:Ray (Order Theory)/Open

## 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:

$\left\{{x \in S: a \prec x}\right\}$ (the strict upper closure of $a$), denoted $a^\succ$
$\left\{{x \in S: x \prec a}\right\}$ (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:

$\left({a \,.\,.\, \to}\right)$ for $a^\succ$
$\left({\gets \,.\,.\, a}\right)$ for $a^\prec$

can also be used.

## Also see

• Results about rays in the context of order theory can be found here.