Definition:Interval/Ordered Set/Closed

Definition
Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $a, b \in S$.

The closed interval between $a$ and $b$ is the set:


 * $\closedint a b := a^\succcurlyeq \cap b^\preccurlyeq = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }$

where:
 * $a^\succcurlyeq$ denotes the upper closure of $a$
 * $b^\preccurlyeq$ denotes the lower closure of $b$.

Integer Interval
When $S$ is the set $\N$ of natural numbers or $\Z$ of integers, then $\closedint m n$ is called an integer interval.

Also defined as
Some sources require that $a \preccurlyeq b$, which ensures that the interval is non-empty.

Also see

 * Definition:Open Interval
 * Definition:Half-Open Interval:
 * Definition:Left Half-Open Interval
 * Definition:Right Half-Open Interval


 * Definition:Closed Real Interval