# Definition:Ray (Order Theory)/Open

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## 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

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

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