# Definition:Ray (Order Theory)/Closed

< Definition:Ray (Order Theory)(Redirected from Definition:Closed Ray)

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## Definition

Let $\struct {S, \preccurlyeq}$ be a totally ordered set.

Let $a \in S$ be any point in $S$.

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

- $\set {x \in S: a \preccurlyeq x}$ (the upper closure of $a$), denoted $a^\succcurlyeq$
- $\set {x \in S: x \preccurlyeq a}$ (the lower closure of $a$), denoted $a^\preccurlyeq$.

## Also known as

A **closed ray** is also sometimes referred to as a **closed half-line**.

The notations:

- $\hointr a \to$ for $a^\succcurlyeq$
- $\hointl \gets a$ for $a^\preccurlyeq$

can also be used.

## Also see

- Results about
**rays**in the context of**Order Theory**can be found**here**.

## Sources

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