Definition:Interval/Ordered Set/Closed

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

Let $a, b \in S$.

Then $\left\{{s \in S: \left({a \preceq s}\right) \land \left({s \preceq b}\right)}\right\} = {\bar\uparrow} a \cap {\bar\downarrow} b$ is called the closed interval between $a$ and $b$.

In the above, ${\bar\uparrow}$ and ${\bar\downarrow}$ represent upper closure and lower closure, respectively.

It is written $\left[{a \,.\,.\, b}\right]$.

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