Definition:Interval/Ordered Set/Closed

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

Let $a, b \in S$.

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


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

where:
 * $a^\succeq$ denotes the upper closure of $a$
 * $b^\preceq$ denotes the lower closure of $b$.

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

Also see

 * Definition:Closed Real Interval