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.

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

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.