# Definition:Ray (Order Theory)/Closed

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

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## Definition

Let $\left({S, \preccurlyeq}\right)$ 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**:

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

- $\left[{a \,.\,.\, \to}\right)$ for $a^\succcurlyeq$
- $\left({\gets \,.\,.\, a}\right]$ for $a^\preccurlyeq$

can also be used.

## Also see

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