Definition:Interval/Ordered Set/Closed

Definition
Let $\left({S, \preccurlyeq}\right)$ be an ordered set.

Let $a, b \in S$.

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


 * $\left[{a \,.\,.\, b}\right] := a^\succcurlyeq \cap b^\preccurlyeq = \left\{{s \in S: \left({a \preccurlyeq s}\right) \land \left({s \preccurlyeq b}\right)}\right\}$

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 $\left[{m \,.\,.\, n}\right]$ 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