\begin{definition}[Definition:Interval/Ordered Set/Closed]
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 $b$.
\end{definition}